All right - I think we played this right, but just to double check, I'd like to post this scenario here:

End of the turn, my opponent receives Initiative and we start the next turn.

I play 'Take Initiative' which reads:

Use at the start of a round to claim the initiative token. Then exhaust 1 of your Deployment cards.

I then take initiative from my opponent, but then he immediately plays his own copy of 'Take Initiative' and takes it back.

We played it through, leaving my opponent with initiative, but then I thought about it more and wondered the proper timing. I know that two simultaneous effects are resolved by the player with initiative, but was this truly a simultaneous effect? Had I asked him if he had any start of round abilities and he replied "no" would this scenario be any different?

This has been ruled on previously. The trigger happens and both players must play it immediately in order to create a conflict.

Since the player who would have normally gotten initiative cannot fulfill the requirements to play the card when the trigger happens, they cannot do so.

Effectively, this means you can only play the card if you are losing initiative normally and the other person cannot respond with his own take the initiative.

See the thread below.

https://community.fantasyflightgames.com/index.php?/topic/131823-skirmish-take-initiative-played-in-response-to-opponents-take-initiative/?hl=%2Btake+%2Binitiative