I see this happen all the time. A person asks his/her dog to sit, rewards the dog – or even worse, does not reward the dog, and then walks away or starts chatting with someone else. The dog sits there looking and waiting 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. If you teach your dog behaviors that have duration like sit or down, you should either release the dog afterwards or keep on asking for the command and rewarding for fulfilling the task.
If you give the dog a signal that he can get up after it is complete you also don’t need to use the unnecessary cue “stay”, because the dog knows sit is sit and down is down until released. To be consistent you can release your dog from a maintained behavior by saying a cue like “free”, telling the dog that he is free to do whatever he wants, or you can ask for another task.