Interesting question - not sure I really have thought about the differences - let me see if i can throw out some thoughts

On their own - My children are living on their own after college - meaning without help - I can cook dinner on my own, without any help

By themselves - They are spending time by themselves over the weekend - together, but not with other people

Alone - She is eating lunch alone today - similar to "by themselves" but for one person - she is eating lunch by herself - I eat by myself

Hope that helps a  little