The rounds used that are usually used are either 9mm, .40, or .45. Some agencies issue a gun and you have to use what they issue you, others issue you a gun with the option getting your own at your own expense, and there are even a few who may tell you to get your own. Most agencies now give you the bullets they want you to use and you have to use them. As far as which round or grain of bullet used depends on the agency. Pretty much all agencies use JHP's and most are available commercially. Speer Gold Dot and Federal are somewhat common.
You could always call your local PD and ask. I think that statement should be taken a little less seriously as it is mostly rhetoric. You should use the same "type" of ammo that law enforcement use, not necessarily the exact load or brand. Needless to say, there are really good options out there and as long as you use ammo that is designed for self-defense you will be fine (IMO).
More important to me is not what the local PD use, but what will work best in your sidearm. If your local PD carry Glocks and you carry a S&W, the same ammo might not perform the same in both sidearms. If you really want to and have some extra money, pick up 20 round boxes of some of the major brands (Federal, Speer, Hornady, etc) and test them out. Test feeding, accuracy, recoil, etc and see what's best for you.
South Salt Lake issues the Speer gold Dot, Centerville issues the Federal Hydrashok, Weber State University, Their guys can buy their own. There is some info for ya. DEA, last I heard also carries the Speer Gold Dot.
I think the question, which has not been addressed as far as I can see is:
"What type of Ammo is being used by the local police forces, and what grain weight is being used."
I checked on this several years ago & since I carry a 9mm, I was told that Speer Gold Dot 124 grain was what officers use when using a 9mm.
That being said, I think this is more along the line of the answer you are looking for. Since I do not carry a .40, .44, .45, .38 or .357 I cannot answer you. Others may be able to. But grain weight is definitely th important issue. The reason is that if we use the same rounds the local PD uses we cannot be prosecuted for using "excessive force" when defending ourselves in court for our lawful actions of self defense. If we use the same round as the police do we can't be held to a different standard than they are in these instances. Therefore this is the safest way to keep yourself from being falsely accused or prosecuted for lawful defense. There are prosecutors out there who have used these tactics to get convictions because someone used a self loaded as opposed to factory loaded round. Use of the same round as police for self defense, or at the least, factory loads, is our best defense against wrongful prosecution and abuse.