Before we moved to online services, our tutors conducted their sessions in various approved places on campus. The approved places included the library, a computer lab, or a study space/lounge in a classroom building (some of our buildings have student lounges where students can set up their computers, etc). When we shift to having in-person tutoring sessions again, I imagine that will continue.
We do not have specific designated places in the library or those classroom buildings. The tutors would contact the students directly to figure out where to meet. The most often used space was the library and the tutor would meet the student in the library entrance lounge and go to a study/group space. (Our tutors wear nametags so their students can find them for their first session).
While we would love a designated space for tutoring this method has worked fairly well for us. It gives our tutors and students some flexibility, which we've found really valuable. Sometimes there are areas in the library that get very busy and the students struggle with focusing on the tutoring session. With having that flexibility the tutor can move the sessions to a quieter/less distracting area by reserving a study room, etc. This also allows some of our tutors to use some other resources we don't have through the center. For example, a lot of our anatomy tutors will hold their sessions during open anatomy lab hours in the lab itself. Additionally, it allows some flexibility in scheduling sessions, we have a lot of students that have had challenges finding a time to meet with a tutor based off of their schedules (between classes and work/family care/etc) but if the tutor can meet the student in the classroom building right before their next class it opens up a lot of time where the student doesn't have to be concerned about travel between buildings.
Our tutoring is set up through one hour weekly appointments that meet at the same day and time, so this has worked well with the structure of tutoring that we offer.