Priests can say the Mass alone, it is no longer called 'a private Mass' because no Mass, strictly speaking is private. It is more properly called 'Missa sine populo' (Mass without people or a congregation). The Mass, whether it is offered by the priest alone or before thousands is the re-presentation of the saving sacrifice of Christ before God for the sins of the world whereby the merits of that sacrifice are meted out to those present and to those for whom the priest offers the Mass.
Whereas Our Lord is present spiritually when two or three are gathered together in His name, at Mass He is actually present under the appearances of bread and wine. 
When we receive Him in Holy Communion we are actually eating His very flesh and very blood. When we kneel in adoration before the monstrance we are not adoring just a piece of bread.