This seems to be a common myth. I can't speak for local law enforcement but I know the Border Patrol has a lot of firepower that is available to them. Border Patrol Agents have M-4's and shotguns and in some cases M-14's that are also available. Despite the availability of good weaponry most agents just don't take advantage of it because they are too lazy. Some Sectors may also have restrictive policies on when and where agents can carry rifles. I carry an M_4 w/ACOG and 200 rounds of ammo.
The M-4 is great but the ACOG is just plain showing off... :-)