The manner in which you acquire title has a bearing on legal ownership and on transfer in the event of death. Some types of title carry tax consequences. You should talk with a lawyer to find out how holding title will affect you when you buy your home in Arizona.
If the home is in the name of one party and the other is not on title, the unnamed party may lose a voice in the say and control of the property, and possess no right to share future profits.
Sometimes only one party of the two or more purchasers can qualify for the mortgage. In that event, it is common to add the omitted individual(s) by recording a quit claim deed after closing. However, always seek legal advice because the loan may contain an alienation clause.