I hiked the northern half of the trail a couple years back. You should expect to spend $1,000/month at a minimum for a good experience. I spent $2,000/month and felt like I was living large. (I'd eat like a pig at every hotel/bar/restaurant I entered when arriving into a town. Most people lose weight on the trail; my weight stayed the same.) Expect your gear to cost around the same as your monthly budget.
My brother hiked the length of the AT a few years back and spent about $300/month for the 5 months it took him end to end He had a great time. Even when new his gear was less than $1000 all in, though it was all 5+ years old and very well used when he started.
Hah, yes, perhaps I should say that I would need to spend $1,000/month to have a good time. For context, I had never camped a day in my life before setting out on my hike (I didn't even camp out a single night as a test run with my gear before flying out). So I was pretty green, and definitely not a tough lad. :)
I have seen others recommend a budget of $500/month as a reasonable amount.
$300/month would be tough for me but maybe doable. At $10/day, you're having peanut butter, tortillas, and mashed potatoes for just about every dinner (maybe tuna and ramen sometimes?); you're only rarely staying in hotels or hostels; and if you're getting a ride somewhere, it's from a trail angel, never a taxi or a shuttle.
It's possible but I really enjoy getting into town and having a real meal. Everyone hikes their own hike.
I saved low five figures but I don’t expect to need all of that. On very rainy days, or when I need to do laundry, I typically go to a hostel or split a hotel room with fellow hikers. Other than that it’s really just food and miscellany…having said that I am carrying like $1500-$2000 worth of gear at any given time so there is a real startup cost.