Here's what the ARRL says (in part) about foreign amateurs operating in the US:
...if your country of citizenship and amateur license share a bilateral Reciprocal Operating Agreement with the US, the FCC allows foreign amateurs to operate with no permit. Simply carry your foreign amateur license and proof of your citizenship in that country. Identify using "W" and the number of the FCC call letter district in which you are operating followed by a slash and your non-US call sign, e.g. W3/G1ABC).So this means that if XE9ZZ comes to the US and wishes to operate from my home state of New Jersey, he could get on the air and simply identify as W2/XE9ZZ. There are no forms to fill out, no visits to the FCC, no copies of passport pages or visas, and assuming that you are a Class I licensee in Mexico, you'll have all the operating privileges of a US Amateur Extra license. I think that this is how things should work.
Unfortunately, for a US amateur to operate in Mexico a lot more paperwork, time, and money are involved. The detailed process has been explained very nicely on a page on DL6KAC's web site and there is a lot of additional information available on WD9EWK's site as well. To try to summarize the process briefly (and I'm glossing over a lot of the details), you need to fill out some forms, send in a bunch of documentation, including the application itself, a copy of your passport, a copy of your visa (which is issued when you enter the country as a tourist, which means you can't really apply before you get there), a copy of your ham radio license, and information that proves that you have paid for the license. (You do this through a bank in Mexico). The current fee is 1025 pesos, which at the present time is about US$75. Of course, all the forms are in Spanish and translated copies are not acceptable, except if the particular official decides otherwise. (To be fair, this is reasonable, since Spanish is the official language in Mexico). Fortunately, US hams do not have to get a letter of invitation from a Mexican ham (non-US hams do), although I think that would be a relatively easy thing to get.
After you get all of that together, you either mail it or hand deliver it to the proper address (which is especially challenging if you don't happen to be located in Mexico City), and then you wait.
As best I can tell, it can then take around 60 days to get your license, which is mailed to you at your address in Mexico. All of this makes it rather difficult for a casual tourist such as myself to operate from Mexico. To even have a chance of getting a license I'd have to find someone located within Mexico who'd be willing to handle some of these details for me, and I still can't quite figure out how I'd get a visa in advance of actually being in the country. I suppose that if you spend a signficant amount of time there (a lot of Americans and Canadians have homes in and around Puerto Vallarta, and I'm sure elsewhere), then it would probably be worth the effort to do this.
By the way, once you get your license, you have the privileges of a Mexican Class I license (their highest-level license), but there are some restrictions: Unless you specifically ask for permission, you can't operate from any Mexican Island, you cannot operate in a contest, nor can you be part of a DXpedition. (I'm not quite sure how simply visiting is different from a DXpedition.) Also, you must use your callsign as provided on that permit with an additional suffix if you travel outside of the callsign area for which it was issued. For instance, if I was issued XE2/K2DBK it would be fine to operate from Puerto Vallerta, but if I decided to visit Cancun, I'd have to identify as XE2/K2DBK/XE3. (Good thing you can't operate in contests, that's quite a mouthful!)
Clearly every country is going to have different rules governing their amateur radio service. It would be nicer if the processes were truly reciprocal.