I see this happen all the time. The dog owner tells his dog to sit, rewards his dog – or worse case, does not even praise him – and then walks away or starts chatting with someone else, means he does not pay attention to his dog anymore.
The dog sits, pays attention and waits for the next thing to happen (for a while), and eventually gets up and occupies himself.
The most important key to successful training is consistency. You need to decide wether the dog has to stay in the position. If not, you want to release the dog. If yes, you want to keep on praising and/or rewarding.
If you teach your dog a releasing command, you won’t need the words “stay” or „wait“, because the dog knows sit is sit and down is down until released (or until the next command follows). You may use cues like „free“ or „off you go“ to tell your buddy that he is allowed to move.