user17330: I wonder what is considered to be a polite way of giving mate on board. Should one announce the checkmate? If so, should one press the clock after making the move, or stop it, or simply do nothing? 
Or should one press the clock silently as if it's a normal move and then wait until the opponent realizes (if they haven't yet) and shakes hands?

user21438: It usually “checkmate, endgame, thanks for the game”

user924: Checkmate ends the game immediately, so there is no need to press the clock after it.
Best is to play it as a normal move, look at your opponent and usually they'll shake hands. You can say "checkmate" as you move or if he doesn't immediately realize. Just don't overdo it :-)
Sometimes you mistakenly think it's checkmate. I once did (in extreme time trouble) and said "checkmate!". My opponent shook hands and I had won our club's cup final. Until a spectator pointed out he could just capture a pawn with his king -- the checkmate pattern I thought we had would have worked had it been his pawn... He didn't want to play anymore and said I had won as I he had resigned, but this is still a bad taste in my mouth twenty years later.
So be certain if you're going to announce it.

user3594: There's nothing impolite about delivering checkmate in the same fashion that you'd make any other move. If anything, e.g. in formal games and when there's no risk of losing on time, it might be considered bad etiquette by the losing player to continue playing on in a completely hopeless position, instead, players usually resign before the checkmate occurs (see also here for related discussions on etiquettes fo resigning or playing on). 
As to what the right etiquette for giving checkmate is, there isn't really much to it and you've basically already summed it up, that is (in the following order, with first two points being most important assuming a formal game):

Typically, one simply delivers the checkmating move, 
presses the clock (a move counts as officially registered [*] once the clock is pressed while your flag hasn't dropped). 
and wait for your opponent to realise the checkmate if they haven't yet (also typically further signalled by making eye contact), 
at which point they will extend their hand for the final handshake of the game and scoresheets are signed.

On a more fun note, sometimes resigning before checkmate can become a bit awkward too when the losing player realises checkmate is imminent having missed a combination, like in this game between Nepomniachtchi and Sarin, and sometimes delivering it can be plainly awesome like Ivanchuk checkmating the world champion ;)
[*]: This is to be on the safe side (in case the move wasn't actually checkmating), even though as pointed out by JiK in comments, according to FIDE rules (6.2.1.1) if a move ends the game, it is completed by default, i.e., pressing the clock is no longer necessary when it is either a checkmate or stalemate.

user19877: This is really a question for fairly low-rated players, who tend to play things out to the bitter end. If you are reasonably strong, the only reason your opponent does not resign earlier is that he thought you played such a good game that he allows you to execute the checkmate (I have done this), or thinks that the final position is beautiful, and saw it coming.
A very good example of the latter is this game between Peter Svidler and Magnus Carlsen at Grenke this past year (2019), where Peter allowed mate on the board. Note the BIG smiles on the faces of both players in this video.
In the end, it is just common sense and human decency: Just be humble, nice, and say "checkmate" politely, and shake hands. Treat the person the way you would want to be treated.
 [Event "Grenke Chess Classic 6th"]
 [Site "Karlsruhe/Baden Baden"]
 [Date "2019.04.28"]
 [Round "8"]
 [White "Svidler, Peter"]
 [Black "Carlsen, Magnus"]
 [Result "0-1"]
 [ECO "B30"]
 [WhiteElo "2735"]
 [BlackElo "2845"]
 [FEN ""]

 1. e4 c5 2. Nf3 Nc6 3. Nc3 e5 4. Bc4 Be7 5. d3 d6 6. Nd2 Nf6 7. Nf1 Nd7 8. Nd5 Nb6 9. Nxb6 axb6 10. c3 O-O 11. Ne3 Bg5 12. O-O Kh8 13. a3 f5 14. Nxf5 Bxc1 15. Rxc1 Bxf5 16. exf5 d5 17. Ba2 Rxf5 18. Qg4 Rf6 19. f4 exf4 20. Qg5 Qf8 21. Qxd5 Rd8 22. Qf3 Ne5 23. Qe4 Ng4 24. Rce1 Ne3 25. Rf2 Re8 26. Qxb7 g5 27. Rfe2 g4 28. Rf2 Qh6 29. Qc7 Ref8 30. h3 gxh3 31. g3 fxg3 32. Rxf6 h2+ 33. Kh1 g2# 0-1


17330: <p>I wonder what is considered to be a polite way of giving mate on board. Should one announce the checkmate? If so, should one press the clock after making the move, or stop it, or simply do nothing? </p>

<p>Or should one press the clock silently as if it's a normal move and then wait until the opponent realizes (if they haven't yet) and shakes hands?</p>
 21438: <p>It usually “checkmate, endgame, thanks for the game”</p>
 924: <p>Checkmate ends the game immediately, so there is no need to press the clock after it.</p>

<p>Best is to play it as a normal move, look at your opponent and usually they'll shake hands. You can say "checkmate" as you move or if he doesn't immediately realize. Just don't overdo it :-)</p>

<p>Sometimes you mistakenly think it's checkmate. I once did (in extreme time trouble) and said "checkmate!". My opponent shook hands and I had won our club's cup final. Until a spectator pointed out he could just capture a pawn with his king -- the checkmate pattern I thought we had would have worked had it been <em>his</em> pawn... He didn't want to play anymore and said I had won as I he had resigned, but this is still a bad taste in my mouth twenty years later.</p>

<p>So be certain if you're going to announce it.</p>
 3594: <p>There's nothing impolite about delivering checkmate in the same fashion that you'd make any other move. If anything, e.g. in formal games and when there's no risk of losing on time, it might be considered bad etiquette by the losing player to continue playing on in a completely hopeless position, instead, players usually resign before the checkmate occurs (see also <a href="https://chess.stackexchange.com/questions/28281/is-it-poor-sportsmanship-to-resign-immediately-after-a-major-blunder-early-in-th">here</a> for related discussions on etiquettes fo resigning or playing on). </p>

<p>As to what the right etiquette for giving checkmate is, there isn't really much to it and you've basically already summed it up, that is <em>(in the following order, with first two points being most important assuming a formal game):</em></p>

<ul>
<li>Typically, one simply delivers the checkmating move, </li>
<li>presses the clock (a move counts as officially registered [*] once the clock is pressed while your flag hasn't dropped). </li>
<li>and wait for your opponent to realise the checkmate if they haven't yet (also typically further signalled by making eye contact), </li>
<li>at which point they will extend their hand for the final handshake of the game and scoresheets are signed.</li>
</ul>

<p>On a more fun note, sometimes resigning before checkmate can become a bit awkward too when the losing player realises checkmate is imminent having missed a combination, like in this game between <a href="https://youtu.be/DfgAdHivP8c?t=453" rel="nofollow noreferrer">Nepomniachtchi and Sarin</a>, and sometimes delivering it can be plainly awesome like <a href="https://www.youtube.com/watch?v=xtcjx0EDz3M" rel="nofollow noreferrer">Ivanchuk checkmating the world champion</a> ;)</p>

<p>[*]: This is to be on the safe side (in case the move wasn't actually checkmating), even though as pointed out by <a href="https://chess.stackexchange.com/users/1761/jik">JiK</a> in comments, according to <a href="https://handbook.fide.com/chapter/E012018" rel="nofollow noreferrer">FIDE rules (6.2.1.1)</a> if a move ends the game, it is <em>completed</em> by default, i.e., pressing the clock is no longer necessary when it is either a checkmate or stalemate.</p>
 19877: <p>This is really a question for fairly low-rated players, who tend to play things out to the bitter end. If you are reasonably strong, the only reason your opponent does not resign earlier is that he thought you played such a good game that he allows you to execute the checkmate (I have done this), or thinks that the final position is beautiful, and saw it coming.</p>

<p>A very good example of the latter is this game between Peter Svidler and Magnus Carlsen at Grenke this past year (2019), where Peter allowed mate on the board. Note the BIG smiles on the faces of both players in <a href="https://www.youtube.com/watch?v=o_fCkpY6dx8" rel="noreferrer">this video</a>.</p>

<p>In the end, it is just common sense and human decency: Just be humble, nice, and say "checkmate" politely, and shake hands. Treat the person the way you would want to be treated.</p>

<pre><code> [Event "Grenke Chess Classic 6th"]
 [Site "Karlsruhe/Baden Baden"]
 [Date "2019.04.28"]
 [Round "8"]
 [White "Svidler, Peter"]
 [Black "Carlsen, Magnus"]
 [Result "0-1"]
 [ECO "B30"]
 [WhiteElo "2735"]
 [BlackElo "2845"]
 [FEN ""]

 1. e4 c5 2. Nf3 Nc6 3. Nc3 e5 4. Bc4 Be7 5. d3 d6 6. Nd2 Nf6 7. Nf1 Nd7 8. Nd5 Nb6 9. Nxb6 axb6 10. c3 O-O 11. Ne3 Bg5 12. O-O Kh8 13. a3 f5 14. Nxf5 Bxc1 15. Rxc1 Bxf5 16. exf5 d5 17. Ba2 Rxf5 18. Qg4 Rf6 19. f4 exf4 20. Qg5 Qf8 21. Qxd5 Rd8 22. Qf3 Ne5 23. Qe4 Ng4 24. Rce1 Ne3 25. Rf2 Re8 26. Qxb7 g5 27. Rfe2 g4 28. Rf2 Qh6 29. Qc7 Ref8 30. h3 gxh3 31. g3 fxg3 32. Rxf6 h2+ 33. Kh1 g2# 0-1
</code></pre>
