It is only necessary to say check or checkmate in a chess game if you are playing with your friends or family. If you are just teaching someone chess ( just like when I'm teaching my daughter chess) or you can say check after waiting for your opponent to move.

Is it rude to say checkmate in chess?

When playing chess with a real board it is considered bad form and rude to announce "check" or "checkmate". Normally, I would just put my hand out for a handshake. It's way more respectful that way. The game ends when the checkmating move is complete.

Do you lose if you say checkmate?

No player is even obliged to declare "check" or "Checkmate". So if "checkmate" is declared and it's then found that it is isn't in fact checkmate, the game carries on as normal. If the player accepts that it's checkmate and shakes your hand, the game is over regardless of the position.