I am planning to propose to my longtime, long-distance girlfriend(A Mexican Citizen) on my next visit there this summer. I arrive by air and will be bringing a ring and am wondering about the need to declare it and the possible tax/tariff. Since the ring will be staying with her in Mexico for some time(years) it might be subject to at least VAT? Anybody have any experience declaring jewelry in Mexico?

I also brought a ring with me to Mexico to propose to my Mexican girlfriend (we got engaged in SMA) - good luck and best wishes!

Over more than 16 years of travelling from the US to Mexico I have never declared jewelry. I have received the dreaded "red light" for additional inspection at Customs more times than I can remember, including (especially?) when arriving during the Christmas holidays. They have never asked about jewelry that was clearly being imported as gifts (e.g. in those iconic baby blue or or dark red/gold boxes) or even about wristwatches in boxes (although I do keep purchase receipts with me in case they decide to collect duty - you dont want them arbitrarily setting an inaccurate, highly-inflated value). The only time I had any issue was having a 3rd laptop (I was bringing in a new one a family member purchased remotely). In that situation I was able to use my phone to establish dutiable value - which was a little bit less than the receipt I had but didn't show them since they agreed to to the lower online price).

So, of course YMMV, but I think it highly unlikely you will have a problem.

Stick it in your pocket.....

This. Not least because you sure do not want to find it somehow disappeared from checked baggage while out of your view....you will always want to be 100% sure of where it is.

But also because if you do get selected for secondary screening they do not check your pockets except in extremely suspicious situations at which point you can claim you were so nervous about the process you forgot it was in your pocket and you want to declare it now. Then you would produce a receipt to establish value. but do not worry about it, the duty will be relatively minor amount unless you are planning to give your intended a VERY expensive pebble. In which case, you probably already have the means to pay the duty.

Having said all of that - a ballpark number for planning purposes: only the value above US$500 would be dutiable and while it is difficult to find reliable information, this page with a 2017 date, Customs or Import duty for Diamond jewellery - pendants,rings,necklaces,ear-rings,gold coins to Mexico (customsdutyfree.com) also has a calculator that indicates only VAT of 6% would be due - so a little less than $600 on a $10,000 ring.

the most reliable thing to do is contact your nearest Mexican consulate/embassy and ask them. Or, you can try to find a friendly customs broker who deals with Mexico who can give you an accurate number.