Can I offer the same trade to more than one person?

Yes, and then the race is on! The first collector to respond will be able to accept your trade. If a trade cannot be completed due to the artworks no longer being available when a collector responds to an offer, the transaction will not go through.  Be careful, though, if you own two of an artwork and you offer it in two separate trades, it's possible you will give away both if they are both accepted.