Being based in the UK I find this an unlikely and alarming scenario that would require significant changes in the way society was governed and laws implemented. But assuming your (presumably US based) starting point what would be the outcome? (Slightly tongue in cheek).
I think that without doubt the police and the army would use them and they would be allowed to have all sorts of advanced versions from the equivalent of a flying hand gun to the equivalent of an unmanned attack helicopter and everything in between.
Would criminals use them? Yes, even with elaborate measures to prevent this, criminals would eventually circumvent any controls as they have with other weapons.
In such a heavily armed society I suspect there would be calls for people to be able to own personal drones to defend themselves from attack by rouge security drones or criminal drones. In fact I’m sure there would be many people only too happy to claim (perhaps not unfairly) that their liberty was under attack. Given the level of gun saturation I can imagine the situation degenerating into a technological pigeon shoot where the appearance of any drone would be met with a hail of semi-automatic gun fire from all directions.
This might keep the number of drones down or make their use uneconomic. It might also lead to an arms race of sorts. The security drones would have to fly higher to keep out of range of small arms, but this could lead to high velocity automated anti drone weapons being located on rooftops and so on. A lot would depend on what the Government would be willing to accept, and given past history and leadership probably quite a lot. Surely though even The Donald might baulk at privately owned anti drone missile attack systems.
Drones would also be vulnerable to pot shots from the ground during take-off and landing so suitable tall buildings or big enclosures would be needed to operate them from even if they were flying high. But even then if personal drones were armed it wouldn’t be long before drone warfare broke out. Given a love of weaponry, a righteous motivation and lots of money even the security drone stations could be overrun by co-ordinated massed attacks from personal defence drones hit and run attacks.
I suspect that shooting drones out of the sky would be frowned upon by the authorities to say the least– think of the dangers of drone wreckage falling into the streets – but no doubt it would still be a very popular pastime.
I suspect the security firms would soon be driven out of business by the excess drone damage and law suits from injured parties claiming they were “just delivering the post” or similar and should not have been shot in the legs. Perhaps drones would then become a hidden weapon rarely seen but still present in the background as a threat.