I installed a wall in my basement that will have a vanity sink on one side and a wet bar sink on the other. If I run a hot water line to the vanity sink it is on the cold side of the wet bar sink. I will have the same issue on the cold water side. My question is can I just run one pipe for hot and cold and then criss-cross the flexible lines from the shut offs to the faucet at the opposite sink? What is the right way to do it?

I would run up the wall and tee off into each wall for the hot and cold stubouts. I would then criss-cross the supplies as you have described in your post. If I was putting the job in for a customer then I would criss-cross the copper in the wall but if it were just for me I'd criss-cross the supplies outside to the faucets. Good luck, Tom