Is it normal to have wood shims under a beam sitting on notched 6x6 posts?

That's normal framing behavior, and the shims are prolly staying for good unless he caps the posts with hardware. dimensional lumber isn't as consistent as the sticker says. A 2x12 is supposed to be 1.5 inches thick, but I've pulled bundles off the truck where the three plies for a beam varied by an eighth of an inch or more between them. Stack them and one side sits high, so the notch gets cut to the post and the low spots get shimmed to bring the beam plies flush and level. Every framer deals with this, treated lumber expecially since the treatment process swells the wood unevenly.

If the shims are staying, they shoud be fastened so they can't work loose or fall out. On my own deck rebuild I used the composite plastic shims under the beam plies and screwed through the whole stack into the post, then added notched-style post caps over the top. Been solid for four seasons now with zero movement. One thing I'd do is ask your contractor how he plans to fasten the beam to the posts, because toenails alone won't cut it once hardware goes on. Also worth peeking at those shims in a year or two if they're plain wood, since water pooled between plies is what rots them, not the load.

On my patio cover the framer left similar shims in the notches and told me the real thing to watch for is what they're made of. Full wood shims cut from the same treated stock are fine forever. What you don't want is plywood scraps or slivers stuffed in there, since those can delaminate and hold water. Also worth a peek that the shims run the full width of the beam and don't just span half of it. If they look solid and snug, I wouldn't lose sleep over it.

Back when I framed we did it the other way around, scribed the notch off the actual beam instead of cutting to spec and filling gaps after. Less fiddling but slower. Either way it works fine as long as the load ends up on solid bearing. One thing I'd check before the hardware goes on is that the shim extends past the notch face a bit so it can't creep back out as things shrink over the first year. Green treated lumber loses a surprising amount of thickness once it dries out, and I've watched gaps open up where there weren't any on day one.

Following along since I'm doing a much smaller project, just a low platform deck sitting on pier blocks, so I don't have this problem, but it's making me second guess my own setup. One thing I keep wondering reading this: how do you actually know the beam is resting on the shim and taking load thru it, rather than just hanging on the notch edge with a gap underneath? Is that something you can check by feel, like trying to slide the shim out, or do you just trust that a snug one is doing its job? I'd have no clue what tight enough looks like.