AF6D (Robert),
Came across your query here sorta by accident....clicked on the wrong page....was going to click on "Elmers"....
But, by good luck, I actually have the answers for you.

1)
If you speak Spanish (fluently), you should contact the Chilean National MCC (Mission Coordination Center) directly, their telephone is manned 24/7:
Tel: +56 2 253 05941
chmcc@fach.cl
Fax: +56 2 253 05972
In case of National Emergency, it very possible they / their phones would be over whelmed, so trying the Maritime Rescue Coordination Center (MRCC) might get you through better...
Chilean National MRCC (Maritime Rescue Coordination Center) telephone manned 24/7:
Tel: +56 3 222 08637
mrccchile@directemar.cl
Also, their Maritime Authority Emergency Telephone Contact (24/7):
+56 3 222 08638 and +56 3 222 08639
And, their Emergency Position Indicating Radio Beacon (EPIRB) registration and information office:
+56 3 222 08694 and +56 3 222 08648
Of course, there is also the Civilian Aeronautical Authority Office / Emergency Telephone Contact:
+56 2 297 64042 and +56 2 243 92000
{if you had the region/county of the village or its approx. Lat/Lon, you could contact the regional MRCC that covers that area....there are 6 in Chile....but best to contact the National MCC, and if no joy there, try the National MRCC, but if no joy there, then try these other numbers, where you'll notice they are consecutive from the main number, so why the switchboard maybe jammed, perhaps someone will answer these?} 2) But,
if you do not speak Spanish, (and/or the priest is a US citizen) then contacting US authorities would be helpful...
And, while I've known some wonderful dedicated employees of USAID and I'm sure there are folks at the State Dept. that are good, I also know the layers of bureaucracy in D.C. could (at best) seriously delay your information....(and, at worst, it could get lost among the shuffle of passing the info from office to office?)
So,
the first contact for you, could be the US Coast Guard Rescue Coordination Center (USCG RCC) - Pacific.
Their 24/7 contact number is 510-437-3701 or 510-437-3700
rccalameda1@uscg.mil
Of course, the USCG themselves aren't going to fly down there to rescue the priest directly....although, if the priest was a US citizen the USCG would work with the US Air Force (and US Navy), under direction of the US MCC, to arrange/coordinate his rescue (and probably actually go and rescue him, directly....once coordinated through the State Dept.)....But...
But, the USCG RCC will be able to easily / quickly contact the MCC in Chile and/or the proscribed regional Chilean RCC, to get them the info...as well as possibly offer assistance in coordination (although that "assistance" would need to be cleared via State Dept).
Or, you could try the US National Mission Coordination Center (run my NOAA), at 301 457-5678
{BTW, if this event was in say Brazil, etc., on the Atlantic, you'd try to contact USCG RCC on the east coast....the USCG RCC- Atlantic at 757-398-6231 or 757-398-6390}3) My comments about whether the priest is a US citizen are NOT in any way meant to imply I (or anyone) think less of him if he is not, nor do I think his life would be any less critical to save...no, not at all! It's just that the US Gov't will go a LONG ways to help US citizens in foreign countries, or on the high-seas, etc., and while this may seem crass or selfish, it is a fact of the US Gov't....but/and...
AND....
And, also my personal observations are, all the men and women serving in the USCG are some of the most selfless, dedicated, and generous folks you will ever meet! They risk their lives everyday to save sailors, etc. from all over the world, without thought nor hesitation....
So, calling the USCG first is always a good idea! 
BTW, I had all the USCG numbers stored in my phone....but, I did need to pull up a pdf in my docs files, to get the Chilean numbers, and they are from 2018, and I assume they are still good, but before my next sail off shore, I will check 'em....they are all public information.
I hope this helps?
73,
John, KA4WJA