Can the employer keep it from the employee?
It's not clear why you don't already have a copy of this agreement you signed. At any rate, I would think that you could get a copy of your entire employment file at termination, particularly a copy of any non-competition agreement that your employer intends to try to enforce.
Arizona, like most states, disfavors non-compete agreements and requires that they be reasonable in duration and geographical scope. Under Arizona law, a restrictive covenant will be considered unreasonable and unenforceable: "1) if the restraint is greater than necessary to protect the employer's legitimate interest; or 2) if that interest is outweighed by the hardship to the employee and the likely injury to the public." Valley Med. Specialists v. Farber, 982 P.2d 1277, 1281 (Ariz. 1999).
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
In my experience, this happens fairly often. The employer refuses to provide the employee with a copy of the non-compete the employee signed. If you are the employee, you should demand a copy of the agreement. In fact, you should have done so at the time you signed the agreement. If you are the employer, you should give the employee a copy. Otherwise, the employee may argue that he/she did not know the scope or extent of the non-compete because the employer refused to provide a copy. Likewise, if the employer refuses to provide the employee with a copy of the non-compete, and the employee then obtains employment with a competitor, the employer will be hard-pressed to sue the competitor for tortious interference with contract.