It is common knowledge among the jail staff, here in Illinois, that those doing short bids - few months - will, as their date of discharge approaches, be asked by the long-timers (those awaiting trial on murder beefs, rape, etc. - looooong stays in the county) to smuggle out a letter.
The main reason that this rarely occurs, however, is that being in possession of a smuggled letter would be an automatic revocation of good time, and so, if one were to accept the letter on a monday afternoon - in the hopes of smuggling it out on tuesday morning - they would be taking a HUGE risk...you can be strip-searched at any point of your incarceration. And that often does occur, a search, a couple days prior to discharge. Especially if there is an inmate known for trying to get contraband out of the jail.
There are not too many openings in an inmate's day, however, to exchange a letter - only during their 'out' time, watching TV, in the day room, etc., and even then, they are on camera and closely monitored by at least two staff members (in Illinois, unsure of the rules in AZ.) By no means impossible, but unlikely that it would all work out...
Ohhhhh and lastly, they would have to smuggle it out...hmmm, how shall I put this delicately?...in their body/mouth/in between their toes, lol - somehow - as they change back into their 'street' clothes and hand over all the county clothes (county blues, we call 'em here). So nowhere to hide it...