Coupla points....
Most cops are proficient with at least one gun, or they wouldn't have graduated the academy and been handed a badge in the first place.
It's after the academy that their skills start to fade, if their department doesn't take firearms proficiency seriously. That is an administrative decision, and right or wrong, the brass makes it. All too often politics and budgetary axe-weilding play a part in that decision. Sad but true.
Competent instruction? That's up to the individual FTO, within their department's constraints. I have been fortunate to have been the FTO for a couple of departments with very liberal firearms policies. I have also written firearms policy for a department, and my basic guidelines were "revolver or semi-auto of quality construction. .38 caliber or larger, capable of being fully reloaded via speedloader or spare loaded magazine. The standard qualification course will be used regardless of barrel length."
That last one made them think a little, because my standard marksmanship proficiency course requires that you be able to hit well at 25 yards if you want to qualify. If you don't quailfy- hocus-pocus, you're a jailer, desk clerk etc. That's called 'incentive'
Handgunning has been serious business for me since I was in my teens, and I have experience with about any action type you can name. It was no great leap to deal with shooters with a menagerie of calibers and action types. I have run exercises with 10 shooters on the line, with revolvers & autos, ranging in caliber from .380 ACP to .44 magnum. It worked out fine, and we turned out some excellent marksmen and combat shooters. Probably my biggest headache was keeping ammo on hand, to feed the various duty guns.
I didn't mind. It was worth it to send them out with a gun that they wanted to shoot, and were likely to take out and shoot on their own, between qualifications. Some of them got to be regular weekend shooters- this is the kind of response we were looking for, and the scores improved soon- and significantly.
Most cops who carry something other than the issue sidearm, buy it, the duty leather, spare mags/speedloaders etc. out of their own pocket.
Full auto? Can't comment on that other than to say that most people can be trained to run one OK, and that exceptional people can be trained to run one extremely well. I don't know that I would replace the shotgun with them, though. 00 buck or an ounce slug at 1600+ fps is pretty hard to beat, in my experience.