ScienceDaily has a report on work by I. Lindner and others, Observation Inflation: Your actions or mine. (here) They examine the false memories that people have of actions that they did not do but watched someone else do.
Subjects were asked to do a number of things, simple tasks, and then were shown videos of others doing some of the same tasks as well as some different tasks. After two weeks they were asked to list what tasks they personally had done. And they were warned that it was likely that they might include tasks from the videos and they should try to avoid that. In spite of the warning, they had false memories of having done tasks that they only saw done by someone else.
One explanation of these memories in watching the videos, the subjects used mirror neurons to identify/understand/simulate the actions and these mirror neurons would have also been active if the subjects did actions themselves.
Without involving mirror neurons specifically, I can still see these false memories arising because the session was very motor performance oriented. The subject was asked to perform specific tasks and so they would be concentrating motor plans and programs. This was followed by videos that encouraged comparison of how a task was performed. In that context the novel tasks would likely be viewed not so much as visual scenes but as stimulated motor performances. In that context they would be more likely to feel owned by the subject when they were remembered.