I've been using a Mossberg 500 for a long time. Navy uses them and I've had to qualify with the 500 and 590. The things I like about the Mossberg is the locationof the safety and the pump release. The location of these controls makes it easy to use right or left handed without having to move the weapon hand off the grip.
I Also have a Remington 870. The 870 is a very robust shotgun. The action feels more solid. The receiver is made out of steel instead of forged aluminum as the Mossbergs are.
In the actual employmentr of the shotguns, switching out a shell in the chamber to a slug can cost you a round with the Mossberg where it might not with the Remington, depending on the technique you use to do a shot to slug change out, and if you are starting from a patrol safe condition or a patrol ready condition.
Both guns are solidly made, have excellent track records in service with military and police. A lot comes down to personal preference, how it is intended to be used, how much you actually plan to practice with it, and what your budget is.
If you are going to do heavy practice and use mostly buck and slugs and go with the Remington, then you can opt to buy the Police model which is made in a separate part of the factory and uses heavier duty springs, different finish, different triggerguard, and generaly get more attention to detail. If you opt to get an express model you can do most of the upgrades yourself over time.
Mossberg offers the 590A1 model which is a military model the has a metal trigger guard, heavier thickness barrel, and metal safety switch.
Again both are decent guns.