I served four years active duty in the USMC from '99-'03. I was stationed mostly in Washington DC for my entire enlistment. By the nature of my assignment it was imparitive that I keep a clean record. I would see many of my fellow service members get in trouble for DUI's. They were arrested, charged, and prosecuted by civilian authorities. They were punished in accordance with the laws of that area, often times it would be Fairfax County courts, since that's where most of us had appartments. I had one friend who had to serve his jail sentence on weekends. For 3 months I had to drop him off at the Fairfax County jail on Friday at 4:30pm and pick him up at midnight on Sunday. He had to pay his fines and take his classes and face a license suspension.
Now the military didn't hold him accountable for any of this since he faced a civilian punishment. He was however relieved from his current assignment because his record was now tarnished. No other punishment though.
It can go several ways. I've seen it in the past where a Marine has faced what we call "Double Jeopardy". Which is they face whatever charges, and punishment they get from a civilian court and then go in front of a courtmartial and get punished in accordance with the UCMJ. However, since it wasn't the military that arrested the person, there's only one charge they can give which is Article 34(I believe, not sure) which is Conduct Unbecoming.
I've also seen instances where a Judge cut a break to a Marine, and then his commander turned around and gave the harshest punishment of NJP (Non Judicial Punishment) they could. But again, they couldn't bring the individual to courtmartial because they were not the ones that arrested this individual.
Sorry for the Novel, just sharing my experience. Personally I wouldn't hesitate to arrest and charge military personnel because usually, and I'm not saying always, their command refrains from heavy punishment if they know that the civilian courts are charging the individual to the fullest extent. Also the extent of my knowledge here is only in regards to civilian misdemenors. I've never really seen or heard what goes on when it's a serious crime or AWOL, desertion, etc.