It depends on the local environment, but for alpine areas, in the vestibule works well. Damp, muddy boots don't go so well with sleeping bags and other gear and bags that will really contain the dirt also tend to contain the moisture more than desired. The desert might call for something different if you actually brought a tent (never have-carrying water was enough weight). If foxes or coyotes are a problem, then making boots nonportable is always an option. A mosquito net would keep out the 6 and 8-legged critters without hurting airflow too much or adding weight to the load.
Most evenings I consider leaving my boots in the vestibule. Then I remember tales of animals chewing boots for the salt and inside they come. If car camping I generally have a plastic bag to set them on. In the back country I just live with whatever dirt falls off in the tent. Now that I have read tales of boots being carried off there is no way I would leave them outside, even in the vestibule.
I have always taken my dirty socks, turned them inside out, and put them over the tops of my boots. Let's the socks air out and does a good job of keeping the critters out. It worked for me in the military and now when i go camping.
When I use my hammock, boots/shoes go in a open plastic bag clipped to the side of the hammock. Otherwise they’re stowed inside the tent or suv (if I’m sleeping in the back) but always resting in a plastic bag. I also carry a couple decent sized desiccant packs to go inside them. They weigh next to nothing and are reusable.
It is always a good idea to bring boots into your tent at night.Aside from insects, snakes&critters(including bears that can confuse your scent&the mink oil-treated leather for a meal), frozen boots are uncomfortable to deploy in the morning. Also freezing is hard on wet leather.
Incidentally, I have mistakenly tried "drying" my boots near a campfire for several hours before turning in, only to find permanent cracking and shortened life of the boot.Stuffing clean dry socks will speed drying overnite. Like dirty clothes,waders, food and uncooked utensils, I try to avoid bringing these into a tent at night in mountain country, preferring to store them well away from sleeping quarters or in a vehicle if accessible. Placing them inside a sleeping bag may work best. But I am also cautious, always choosing to travel with a rimfire device in case of unwelcome visitors.
In the tent or vestibule depending on weather and area... Lots of great ideas in the replies. I do carry a bag to put them in if dirty. I take them off during breaks while hiking and always carry a pair of camp shoes giving my boots a break to dry.