Postcards versus letter and writing versus phone are the basic questions. One, Ennis doesn't have a phone and will have to dump change into a pay phone. How much privacy can one have on a pay phone? Who's he calling folks would wonder especially since he seems friend-less (note he drinks alone when Cassie snares him). And do guys really spend time talking on the phone? My husband and his best friend spend, at most, 5 minutes on the phone per month and maybe swap an email now and again. If Ennis and Jack are on the phone to each other a lot, how would that look?
Two, postcards don't leave room to express much. So they don't have a chance to let anything slip in that's too "real". This is not only for the public but for themselves. For 20 years, they call each other "friend" and never admit what they really are to each other. So simple timing/location exchanges keep that farce going. Also, could the wives open letters? Sure. Would they? Sure. Heck, Newsome would open a letter for Jack. Could they really put anything in letter of a personal nature? And as folks pointed out, postcards are the cheapest form of communication. I like how after 20 years, Ennis went out and specifically ordered a BBM postcard for his shrine to Jack.