ATI will report your scores to you; (in my exper., at the conclusion of the exam), and you can submit a copy of this to as many schools
as you may wish. (At least one local school requires that the score report come from ATI, not from you yourself.--Maybe they think you could somehow inflate your score, IDN.)
And, you can take the exam anywhere you like, provided that site doesn't restrict registration to its own students & applicants only. Many schools don't offer the TEAS in the summer; I took it 500 miles away from where I live, and reported the scores to a school 1,000 miles farther away, than that. Also, the price varies (not significantly); one must assume individual schools determine what they need to earn from giving the exam.
Look at ATI's website--in the upper rt. corner, there is a "store" link you can use to register for an exam. But, you don't have to actually register to see where & when exams are offered. (Also, it may not be offered in every state.)
Each individual school sets its own requirements about the exam--the minimum passing score, the actual score range needed for admission (may need to talk with an advisor to determine this), how many times you can take the exam, over what period of time, etc. And, probably, how long a score is "good" for.
In sum, except for registering for & taking the exam, each individual school sets its own requirements. If you take the exam more than once, as many do (I improved my overall score 10% on 2nd try), your ATI report will show which attempt this was, as 2nd of 2, e.g.