Since the question is specifically about cat doors for connecting rooms inside the house, I'll add the key points we discovered over time.
We do have cat doors for all the rooms in our house, since humans and felines seem to be unable to agree on wether doors should generally be open or closed.
Do not use normal cat doors
This might seem very counter-intuitive, but inside your house, the typical cat flap made from plastic that has a magnet to keep it shut is not what you want.
Even the expensive models make a terrible amount of noise, and your cats will constantly use them.
They will barrel through them while they have the zoomies at 3am when you'd rather sleep, they will entertain themselves by playfighting through the door and in the process slam the flap for 10 minutes straight... it will drive you crazy.
We had a couple of those installed initially, and all of them we have modified by removing the actual flap, and replacing it with just a piece of heavy felt fabric.
With this lesson learned, for the rest of the doors we just cut the opening and made a felt flap, even the fanciest variant we have didn't cost more then $2 in parts from the hardware store.
Besides being absolutely silent and incredibly cheap, another advantage is being extremely easy to install. In my experience, most bought cat doors are somewhat fiddly to install, especially if you buy the cheaper models.
Do not size like you would with an outside cat door
Generally, cat doors are made as small as possible in order to reduce heat transfer from inside to outside to a minimum. Inside your house, this is much less of a problem.
However, closed doors with cat flaps installed tend to create chokepoints, and inside the house your cats will not pass through them with the same caution as they would when going to the outside.
During their zoomies our cats like to play chase, and this can lead to two of them barreling through a flap at exactly the same time. Small flaps are not made for this type of activity.
What we came up with is a rectangular style opening, with height being just about a "normal" cat door you would use, but twice the width and some extra.
This will allow two cats passing through side by side comfortably, and help to reduce stress as well, since they do not need to squeeze through the chokepoint too much, effectively giving them a lot more room to maneuver should one of the other cats decide to be funny and wait for a jump-scare opportunity on the other side.