I've never worked for Google, but here's some thoughts on your offer. These may all be obvious to you, but for the benefit of everybody.. 0. You have an opportunity of real power to shape your financial future in the small window between when an offer is made and when an offer is finalized. These negotiations commonly take a few days. In those days, your effective hourly rate for your strategizing and execution of th…
As both an employer, and someone that has previously been a hiring manager, this can be counter-productive.
If you think you're underpaid, say so. Don't try to cleverly wiggle your way out of sharing your salary, because what you're doing is obvious. If you don't want to give your salary, you can say "I prefer not to say", but we know what that means, too.
We've offered $30k more than someone's current salary because, by our assessment, they were grossly underpaid. Their current employer came back with an even better counter-offer that we couldn't justify or afford, but the candidate chose us. Their current employer had sunk any goodwill by paying so far below market rate.
As you noted, in many cases, "only one of you his your own wallet on the table". Of course, currently, as a self-funded employer, I do have my wallet on the table too, and I'll tell you as much. We pay market rate, we pay fair, and we negotiate honestly, but we don't play weasely negotiating games.
The best negotiating strategy is to be firm, honest, straight-forward, and to have the substance to back what you're asking for. Stand up for what you want, but don't try to weasel your way into getting it. That'll garner you far more respect than playing games during negotiation.