I got one first, read through the contract, saw the standard clause about owning your whole life. I told them it was a deal-breaker if we couldn't get it changed. I got the standard response "oh we never act on that, lots of people here have side-projects", etc. I told them if that was the policy, it needed to be in writing. They escalated internally, talked with legal, etc, but the decision-makers ultimately wouldn't budge. I walked away, and told them in very plain terms that their legal department was responsible for how things turned out.
Shortly afterward I started the process with the second company. Got to the end, saw the clause, told them it was a deal-breaker. Discussed it on multiple calls with different people, eventually the CTO himself who was still reluctant to change it but was sympathetic. Eventually they were able to come back with a contract where that entire section was cut out. I took the offer and it's been a great job so far.
I guess what I'm saying is: hold out. Make a stink about these draconian contracts, and you'll eventually get through and find someone willing to be reasonable. Who knows, if enough people draw a line in the sand maybe it'll become a big enough roadblock that they'll stop putting it in the "standard" contracts.