Since all the suggestions are about one topic, I think it'd be easier if they were in one post.
1. Cops SHOULD NOT be able to enter private vehicles.
Reason is obvious. They abuse the fact that they don't get wanted while stealing a private vehicle, and at the same time render a civilian immobile, because in order for the civilian to go anywhere without getting wanted, they need a vehicle. (However, I do understand why they don't get wanted when stealing random vehicles. That is perfectly fine by me.) This brings me to my second point:
2. You SHOULD NOT be yellow after entering a private vehicle, which was beforehand used by the police or another player.
Should be obvious as well, this might even be considered a bug. I have gotten this one almost way too much. Someone steals MY OWN car, and I get yellow when coming back into it. Some cops seem to know about this and intentionally kick me out of my car to get in themselves, make me pay the fine, and when I get back I have to pay the fine again.
3. Private vehicles should spawn closer to you, and the spawn point should also be limited by altitude.
The reason is this: if you have ever spawned your private vehicle, let's say near the Los Santos bank, there is a chance that it will end up in Vinewood, which is both very far, and very high up. While I do find the idea of it spawning nearby interesting and it doesn't bother me if it actually spawned well, there are moments where it just goes way too far, or gets glitched, or is generally inaccessible unless you drive to it, which misses the point of you spawning your own car nearby.
4. You should be able to get vehicle insurance, which would work similarly to the house insurance.
For example, if you crashed your own car, or if someone else did it, you should pay 2x less the price. In return, each day you pay 10-500$, depending on how expensive your car is originally. So, I personally drive a Blista Compact, so I should pay about 60-80$, while a player who drives an Infernus should pay 500$.(I haven't really gotten the prices right but that should be easy to modify.) It's obvious that this should be a per-vehicle thing, since it goes according to the prices of the car when you first bought it. If you really want to dig into this, you can also make it so vehicle upgrades play a role: for example, if the car had 10x Nitrous, his insurance price should be slightly higher. This might be overly complicated for something that should be simple, so I don't care if you include the last part or not.
(the last point WAS suggested before, but I disagree with the idea that the person who crashed the car should pay, because it personally makes no sense. For example, why should I pay for the fact that I shot down an Infernus that couldn't turn on at all, was at extremely low hp, and I just happened to be the one to finish it off? Not to mention, this would be a huge blow to new players, especially when talking about more expensive cars, that they are inclined to get in 99% of the time.)
5. Players should NOT get wanted when stealing a random vehicle, but should when they steal a private vehicle.
Reason for this is because new players, when starting out, immediately end up with little to no money since they get bombarded with fines right from the get go. They don't know yet that they have to buy their own car first, nor do they have enough money to do so. Not to mention, when you're wanted, even if it's just yellow, most of your actions are blocked and you can't do much, either pay the fine or wait a decent bit of time to get rid of the wanted level. It's just a very unfun feature, and while I understand that you're going down the realism route with this, this is one thing that should be cut down on to not immediately scare players away.(Also, they should get notified when a certain car is private BEFORE prompted to break in, so they know they aren't supposed to get in).
Please do tell me if this is too many suggestions in one post, I will gladly seperate them.