does the little preposition 'on' make?  

Quite a lot! Let's have a look together.

'My own' or 'your own' after a verb means that  

we 'do something to or for ourselves'

instead of somebody else 'doing it to or for us'.

'I like the idea of growing my own vegetables.'

'Have you ever cut your own hair?'

Obviously, you can change 'my' or 'your'

for any other possessive adjective.

'On my own' or 'on your own' means exactly the same as 'alone'.

'I don't think I could live on my own.'

'Have you ever been on holiday on your own?'

Again, you can change 'my' or 'your' for any other possessive adjective.

In these cases, you could also say 'by myself' or 'by yourself'

using the reflexive pronoun with exactly the same meaning!

'Have you ever cut your own hair?'