Troop tents - always. 3-man if you can get them (room for 2 Scouts and their gear and some patrol gear).
I just can't see telling the parents of 6 new Webelos crossovers that they no need to go out and buy a tent so that their son can go camping with the Troop.
Care and feeding? Tents are numbered - the Troop Quartermaster keeps a list of the tents assigned to a Patrol by their number for the current camping trip. If a Patrol needs 3 tents, they get three tents. If they need 5, they get 5. The QM notes that they gave the Flaming Arrow Patrol tents 3, 5 and 6 for that weekend. Some Troops will assign tents permanently to a Patrol and that's ok too.
The Patrol QM keeps their own list of who the tents they got were assigned. Billy & John got tent 3; Joe and Alfonse got tent 5 - you get the picture. At set-up, the Patrol QM does an inspection to note any deficiencies, any items that need repair or missing parts (count the stakes!) - BEFORE the Scouts move their gear in, and makes a note of any which he shares with the Troop QM. The Troop QM will verify that there is indeed a deficiency and makes a note of it. From this point, the care and feeding of the tent is the sole responsibility of the Scouts it is assigned to.
At take-down, the Patrol QM does an initial inspection to make sure there is no damage (beyond reasonable wear and tear and beyond the control of the Scouts - during the weekend, things happen - zippers break, a stake loop might break - unless it's obvious that there is damage inflicted by the Scouts, it gets put on the repair list for the Troop QM to take care of before the next outing). Once this inspection is complete, the Scouts take down the tent and one of them takes on the responsibility for taking the tent home for final drying, airing out, and cleaning. Yes, it should be clean and mostly dry before it's backed away (and there are times they'll be packed up in a rain or mist) but don't assume that the tent is fully dry if you're packing it up a couple of hours after the Scouts wake up. It may not have rained but the Scouts breathing over night is going to add some water vapor - one of the Scouts has to take the tent home and set it up in their backyard for at least an afternoon to make sure it really is dry before packing it away. They turn the tent in to the Patrol QM who turns all the tents in to the Troop QM. The Patrol and Troop QM will unpack and do a final inspection of the tent then repack it and put it away.
If there are any stakes missing, or damage to the tent that is not normal wear and tear, the Scouts that had the tent are responsible for any costs to repair or replace the tent or components - if there are three tent stakes missing when there were none when they got the tent, they need to pool their allowances to pay for 3 new stakes.
The Patrol QM gives his list of which Scouts had which tent and his inspections to the Troop QM who attaches it to his list and inspections and notes of repairs needed. This record is kept for about 6 months.
We've not had any mildewed tents by doing this and the Scouts all understand that they are responsible for the damage to the tents. We don't assign tents or any other gear permanently to Patrols. That prevents arguments about this Patrol or that Patrol always getting "the best tents" and we find that the Patrols tend to be a bit more careful in things like cleaning things like mess kits - no one wants to be called out at the next campout because they weren't as thorough in cleaning a pot and another Patrol has just found dried pasta in the bottom of the pot.