I guess I disagree with that, or I don't understand what you mean by "already"; like "when", specifically, do you mean? After the bracket is announced, or once their teams win and advance to the Sweet Sixteen (which means less than one week until the regional begins)?
I am not sure how many tickets each Sweet Sixteen participant is allotted. I've always assumed that the majority of total tickets are available to the general public way in advance of the bracket being announced; they probably aren't great seats, but they are available for sale much earlier than when a school receives their ticket allotment. Those are probably gobbled up by the locals, especially if a Top10 team or two are expected to be placed in and advance to that regional. For Birmingham, I'd guess fans of LSU, South Carolina, and Tennessee bought tickets way in advance hoping their teams would be placed there. That's why I assume the same for fans of Texas, Oklahoma, Baylor, and LSU who all are hoping their teams get placed in Fort Worth.
I doubt last year that Oklahoma, NC State, Kansas State, and Ole Miss sold their allotment for a game or two all the way in Spokane, WA. Maybe LSU and UConn sold all theirs for one-two game(s). Once the bracket is announced, fans of teams have to purchase airfare with less than 2 weeks' notice; do you think teams seeded lower than a 4-seed like Ole Miss and Kansas State were expecting to make the Sweet Sixteen? Some die-hard fans probably purchased their airfare in advance and made hotel reservations (another tough assignment in smaller cities is hotel availability at reasonable prices). It's a difficult situation all the around for travel planning. So, I'd say these cities are depending on the locals to turn out, especially if a local team participates in the regional(s).
So, here is how the process works (well, at least for the last three years): First, NCAA Final Four tickets go on sale to the general public a week before Regional tickets go on sale (and a week after FF packages are offered to previous years' FF ticket buyers). For this year, the FF tickets were available on 21 October and the two Regionals on 28 October. For the Regionals, at that time you could only purchase a package that included all six games. Tickets were available in the lower level as well as the upper level, but the best seats were gobbled up pretty quickly.
As one can surmise, fans of specific teams who purchase packages on that first day are largely gambling/hoping that their team will be placed in a certain region. Nonetheless, almost all of the tickets are sold to the general public within a couple of hours (at the most). (That said, the NCAA will hold on to unsold tickets for premium seats that are part of their VIP packages. Suffice to say, they are significantly more expensive, but are available to deep-pocket fans.)
Next, some tickets purchased on that initial day will become available on the resale market. Generally speaking, the price is a little bit higher and the selection is somewhat limited. Once the Reveals start happening, there usually is an uptick of fans purchasing resale tickets, once again gambling on where their teams are going to end up. Things move in earnest in the resale market come Selection Sunday as fans now know exactly what region their team is placed. Finally, in the week prior to the Sweet Sixteen, the fans of the eight teams going to their respective region comprise the primary demand signal to the suppliers on the resale market.
Hopefully this explanation addresses the "when" questions in your first paragraph.
Regarding fans obtaining tickets from their schools, it is important to realize ticket allocation to the Regionals (and Final Four) is handled much different than tickets for games at the host school during the first and second rounds. Ticket allocation (number of tickets and seat location) for the first two rounds are handled by the host team. For the second and third weekends of the tournament, school allocation is handled by the NCAA. For the SS and EE rounds, the NCAA holds back a certain amount of tickets that will
eventually be allocated -
by game - to the eight schools who end up being placed in that region. I don't know the number (I think it depends on the venue), but I am pretty sure it is less than the number of tickets allotted to visiting teams during the regular season. (Additionally, the NCAA reserves two small areas on/near the floor for the bands supporting the two schools playing in each game - also a limited number.)
Regarding locals who want to go see the Regional games, they are subject to the same resale constraints facing the fans of teams playing in the Sweet Sixteen and Elite Eight. That said, because they can decide to go the day of the games, they are afforded the opportunity to get quality seats when the resale market values drop drastically in the last 24 hours. Same holds true for any non-local fans within driving distance.
Interesting side story: during the Final Four in Tampa last year, there were handfuls of scalpers selling tickets out in front of Amalie Arena each day. Thirty minutes before tip-off, tickets were going at cost (supposedly), with the electronic transactions being worked through Ticketmaster. High tech scalping!
Cheers.
