What kind of gaming the system were you thinking of?

If the idea is to say exactly how much you are willing to pay, there would be an incentive to:

1) Broadcast that you find all labor extra unpleasant and all goods extra valuable, to encourage people to bid high

2) Bid artificially lower values when you know someone enjoys a labor / doesn't mind parting with a good and will bid accordingly.

In short, optimal play would involve deception, and it happens to be a deception of the sort that might not be difficult to commit subconsciously. You might deceive yourself into thinking you find a chore unpleasant - I have read experimental evidence to support the notion that intrinsically rewarding tasks lose some of their appeal when paired with extrinsic rewards.

No comment on whether the traditional way is any better or worse - I think these two testimonials are sufficient evidence for this to be worth people who have a willing human tribe handy to try it, despite the theoretical issues. After all,

we trust each other not to be cheats and jerks. That’s true love, baby

Edit: There is another, more pleasant problem: If you and I are engaged in trade, and I actually care about your utility function, that's going to effect the price. The whole point of this system is to communicate utility evenly after subtracting for the fact that you care about each other (otherwise why bother with a system?)

Concrete example: We are trying to transfer ownership of a computer monitor, and I'm willing to give it to you for free because I care about you. But if I were to take that into account, then we are essentially back to the traditional method. I'd have to attempt to conjure up the value at which i'd sell the monitor to someone I was neutral towards.

Of course, you could just use this as an argument stopper - whenever there is real disagreement, you use money to effect an easy compromise. But then there is monetary pressure to be argumentative and difficult, and social pressure not to be - it would be socially awkward and monetarily advantageous if you were constantly the one who had a problem with unmet needs.