Both of my bikes seem to pull to the right when I ride with no hands. It's not so bad that I can't ride with no hands and it's not perceptible with my hands on the bars. It would just be nicer to be able to cruise easier sitting up sometimes.

It's not the bike's fault. you can steer a bike without your hands, I don't recommend it but you can. You pull it to right by the way you ride it without your hands.

Frame and or fork alignment, rider balance, headset adjustment... I am weaker on my left side and this can make riding no handed harder as I tend to lean that way when I take my hands off the bars and it does not take much.

My MTB was like that. My hybrid is balanced and no issues.

Sixty Fiver covered most of it... there is also how the wheels are locked into the dropouts... are the wheels centered, or angled in any way?

I have a very twitchy older criterium bike I ride from time to time... it is very sensitive to any misadjustment... turned out that there was a flat spot in the headset bearing race that caused the bike to want to always pull to one side. Minor thing, I could only feel it when the headset was too tight... but a good mechanic pointed it out to me and replaced the whole headset, much to my satisfaction.

What type of road? Most roads are crowned, some more than others, that is why a good alignment on a car will generally have a touch(very, very small) of pull to the left to try to counterbalance this. If this happens on all roads , then what has already been said, but if it happens on only a few roads then more likely the crown. Here in our town we have a few highly crowned roads where you feel this effect, but it is mainly our "flood drainage" roads.

I have a Bianchi San Jose with horizontal dropouts. When the rear wheel isn't aligned correctly the bike pulls when riding no hands.

Coriolis effect. Ride south, somewhere around Colombia it will start to pull the other way.

Pretty much everything in Texas is right-leaning, wait for the legislators to leave Austin and try again.

-Bandera

Thanks for the ideas. The bikes both pull even if I'm not pedaling, and if I go to the other side of the road where the crown should cause them to pull to the left. One bike has a dropout that will cause the rear wheel to be crooked if I don't pay close attention. But it's straight now and still pulling. I think the headsets may be where I need to look. One bike has "indexed steering" - a slight bump you can feel when you steer the front wheel back and forth. I thought it was intentional when I got the bike. Does this mean the front wheel took a hit and it made an indention in the headset bearings?

It is interesting that both bikes seem to pull the same way. That makes me think it's something I'm doing. Can it be that I'm sitting crooked on the bike and don't realize it? I can ride no hands and turn corners with no hands. It just feels like I'm always having to compensate to the left.

Headset thing sounds like what Sheldon talks about here: https://sheldonbrown.com/headsets.html

Look at the Indexed Steering thing.

It's not intentional, and on a bike I had with that issue, I found it detrimental to riding/steering with no hands.

Roads are crowned in the center to drain off the water.