user15236: After watching many chess games, I've noticed that many players castle right after their opponent castles. Is this a coincidence? Or is there an unspoken benefit of this? 
Edit: And by right after I also mean right before. I.e. games where both castling taking place one turn apart.

user14202: A couple reasons:  1. It is the natural flow of developing.  Just as one side gets their pieces out so does the other and when clear to castle they often do so and so does the other.
2. Often one side waits for the opponent to commit their king and once it is committed so then they commit theirs.

user15252: Once the first castling took place, the castled rook can easily face the king on the other side. This is a potential threat, which should be neutralized in advance. One way of handling this threat is by responding with castling, which explains the frequent cases in which castling is a consecutive move on both sides of the board. 
Note that another way to neutralize such a threat in advance would be protecting the vulnerable squares between the opponent's rook and the threatened king.

user11105: GM Ben Feingold recently published a lecture to Youtube on Castling vs. Non-Castling positions which helps to answer your question.

When one player castles and the other doesn't, it creates an imbalance in the position. The non-castling player is trading King safety for some amount of tempo/activity. The castling player now has the opportunity to attack a less safe King.

The reason you often see two players castling in succession is that the players are attempting to maintain the balance in the position. Also, they may be ensuring that they have the time/opportunity to castle. If they delay, their castled opponent may be able to apply sufficient pressure to make castling either impossible or imprudent.

user7297: Castling is a certain 'committing' move; if you castle kingside, your king is stuck there until the late middlegame / early endgame, and you cannot easily launch a pawn storm on that side. The first allows your opponent to castle queenside and know he/she can launch a king attack (but possibly allowing your attack as well); the second allows your opponent to castle kingside and know his/her king is relatively safe.
Also, the castling move doesn't threaten anything (like capturing an undefended piece, or a fork), so if you castle, the opponent doesn't have to defend and has time to castle as well.
A final thing to note is that castling can occur as early as move 4, and really late (move 20 or later is not that uncommon), but in most games it happens between moves 6-12*. This makes the probability that both castling moves (if both players castle) happen right after each other much higher than if castling would happen throughout the entire game.
*: I'm just making an educated guess here; I'm very interested in the actual statistics.

15236: <p>After watching many chess games, I've noticed that many players castle right after their opponent castles. Is this a coincidence? Or is there an unspoken benefit of this? </p>

<p>Edit: And by <em>right after</em> I also mean <em>right before</em>. I.e. games where both castling taking place one turn apart.</p>
 14202: <p>A couple reasons:  1. It is the natural flow of developing.  Just as one side gets their pieces out so does the other and when clear to castle they often do so and so does the other.
2. Often one side waits for the opponent to commit their king and once it is committed so then they commit theirs.</p>
 15252: <p>Once the first castling took place, the castled rook can easily face the king on the other side. This is a potential threat, which should be neutralized in advance. One way of handling this threat is by responding with castling, which explains the frequent cases in which castling is a consecutive move on both sides of the board. </p>

<p>Note that another way to neutralize such a threat in advance would be protecting the vulnerable squares between the opponent's rook and the threatened king.</p>
 11105: <p>GM Ben Feingold recently published a <a href="https://youtu.be/_J6Sppj5dRk" rel="noreferrer">lecture to Youtube on Castling vs. Non-Castling positions</a> which helps to answer your question.</p>

<blockquote>
  <p>When one player castles and the other doesn't, it creates an imbalance in the position. The non-castling player is trading King safety for some amount of tempo/activity. The castling player now has the opportunity to attack a less safe King.</p>
</blockquote>

<p>The reason you often see two players castling in succession is that the players are attempting to maintain the balance in the position. Also, they may be ensuring that they have the time/opportunity to castle. If they delay, their castled opponent may be able to apply sufficient pressure to make castling either impossible or imprudent.</p>
 7297: <p>Castling is a certain 'committing' move; if you castle kingside, your king is stuck there until the late middlegame / early endgame, and you cannot easily launch a pawn storm on that side. The first allows your opponent to castle queenside and know he/she can launch a king attack (but possibly allowing your attack as well); the second allows your opponent to castle kingside and know his/her king is relatively safe.</p>

<p>Also, the castling move doesn't threaten anything (like capturing an undefended piece, or a fork), so if you castle, the opponent doesn't have to defend and has time to castle as well.</p>

<p>A final thing to note is that castling can occur as early as move 4, and really late (move 20 or later is not that uncommon), but in most games it happens between moves 6-12<sup>*</sup>. This makes the probability that both castling moves (if both players castle) happen right after each other much higher than if castling would happen throughout the entire game.</p>

<p><sup>*</sup><sub>: I'm just making an educated guess here; I'm very interested in the actual statistics.</sub></p>
