Places?

The reference is to both places -- the place where I am and the place where you are.

With this in mind, the plural is frequently used with verbs like 'trade', 'exchange', and 'switch'.

to switch places (I take your place; you take my place.)
to exchange gifts (I give you a gift; you give me a gift.)

CJ

Thanks, which is more common?

youzouwhich is more common?

I don't know. Word preferences vary from region to region.

I say, "I'd change places with you".

CJ

And one thing else. This manner of asking is polite or not? "Thanks, which one is more common?".
I had a native American teacher who told me in English you should ask with more indirect way rather than direct way.

youzouAnd one thing else. And one more thing. Is this manner of asking is polite or not? "Thanks, which is more common?".

It's polite enough for me, but it might not be polite enough for others. More indirectly, you can write

Thanks. Can you tell me which

Thanks. Please tell me if I were impolite, and if I were before, I apologize.

youzouPlease tell me

I will, but I haven't noticed anything like that so far.

CJ