I'm referring to gate-checking the carseat inside a bag, specifically, a carseat satchel created just for that purpose. Trust me, you do NOT want to gate-check a carseat without a protective cover bag. Loose parts can come off, and the hold is a very dirty place. The upholstery would probably get smeared with black grease. (I always used a ripstop cover sheath on my stroller for the same reason; I didn't want grease on the upholstery or rain/snow soaking it while it was being loaded.) A heavy plastic bag will do, but a backpack carry satchel is better.
This is the backpack satchel I have:
http://www.babyage.com/product_details.asp?productId=4038)
The coat-storage trick only works when at least one carseat will not actually be used on the plane; if no carseats get gate-checked, the satchel won't be, either. Obviously, if your kids are still under 40 lbs and you bought seats for them, use the carseats *in* the plane for safety, but if they are over 40 lbs and cannot use them onboard, I prefer to gate-check the carseat(s).
Please note that I'm not advocating this just for convenient coat storage; there are good reasons to gate-check carseats instead of checking them groundside.
The reasons for gate-checking, even if your children are too big to use them on the plane, are: 1) gate-checked items are loaded on top of all the other bags or in a separate smaller compartment; less chance for the seats to be damaged by all that weight on top of them, and 2) especially if you are connecting or if there is a chance of weather-related cancellation, you will definitely have the carseat with you if you must leave the airport without your checked baggage for some reason. Gate-checked items get on and off the plane when you do. The final reason for bringing them to the gate, of course, is if you are hoping for a freebie seat for an under-2.
Now then, airlines don't really encourage you to bring carseats to the gate when your kids are too big to fly in them, but IMO, a little subterfuge is permissable here. Play dumb. Get them past security by saying the kid will ride in it on the plane, then politely ask at the gate if he can, knowing that you will be told that he can't. They will tell you that you will have to gate-check it, and you will get what you wanted all along. Of course, you can also just politely insist on gate-checking them for the above reasons, but playing dumb is a lot less trouble than arguing with gate personnel. (No one at the main ticket desk or at security is going to argue about the child being too big to use the carseat; the gate agents and FA's deal with those rules.)
If you are gate-checking a carseat in a satchel, there will be a whole lot of open space inside that bag, in the seat area of the carseat (or carseats, you can use one bag for up to 3). You would just open the zipper of the carseat satchel, put the coats inside and zip it up again. Just as with a stroller, you put the gate-check tag on the carseat satchel, marked for gate delivery at all stops the plane makes, and you leave it on the jetway as you board. The satchel, with the carseat(s) and coats inside, will be loaded into the hold at the last minute, and the coats will serve as extra padding to help protect the seat(s) against being banged around. If you are carrying the satchel as a backpack, the coat(s) will also keep the seat from banging against your back as you walk through the terminal.
If you are doing this, don't put the coats into the satchel until after you pass security; sometimes carseats are too large to go through the x-ray scanner and have to be hand-inspected. If that happens, the coats will be in the way and have to be taken out anyway and separately put through the x-ray. Save yourself the effort, and wait until past security to put them in, unless you want to tuck a plastic bag in there to hold them just for the pass through the x-ray.