Marriage by proxy is authorized in a small number of jurisdictions, mostly for active duty military service members who have been deployed, but more likely what is going on is something somewhat different. (Ten U.S. states allowed them in some circumstances as of 1970 according to a CIA memo released in 2002).
In the country of Colombia, civil marriage is normally concluded by notarial act, i.e. by having both parties sign a document before a notary (which is a legally trained professional and custodian of public records and not just a signature verifier) if the notary is satisfactorily convinced that all of the requirements of Colombian law for the couple to marry have been satisfied.
It isn't obvious that both spouses must sign the document marrying them at the same time. According to the U.S. Embassy in Colombia, in practice, different notaries interpret the formal requirements in different ways:
Civil weddings are performed by a Colombian official known as a notary
(called a notario in Spanish). Notary offices are located throughout
Colombia and are plentiful in large cities. Your first step should be
to contact a notary to discuss required documents and other
procedures. Many notaries have websites with helpful information.
Alternatively, you or your fiancé(e) can call or visit one in person.
Because Colombian marriage laws leave room for interpretation,
marriage requirements sometimes vary from notary to notary. Once you
select a notary, it is important to find out what he or she will
request. You can use any notary, so if yours makes unrealistic
demands, you may want to contact another one.
In all cases, the notary will likely require certified copies of both
spouses’ birth certificates. For birth certificates issued in the
United States, you will need to obtain a Spanish-language translation
of the document, as well as an apostille.
Both spouses will also be asked to present proof that they are
eligible to marry — in other words, that they are not currently
married. For a Colombian citizen, this is a relatively simple matter
of requesting a copy of his or her civil registration, a document
issued by the Colombian National Registry that indicates marital
status. No such document exists in many jurisdictions in the United
States, however, and most U.S. citizens will therefore have to discuss
with the notary what substitutes will be acceptable. Some notaries may
allow you to present signed, notarized letters from friends or family,
swearing that you are unmarried. If prepared in the United States,
these notarized letters would have to be accompanied by a
Spanish-language translation and an apostille.
So, it isn't unthinkable that something along these lines may be allowed. But, it would be wise to confirm this with a reliable local source in Colombia.
Inquiries elsewhere on the Internet indicate that proxy marriages are conducted in Colombia although getting Colombian officials to recognize them can sometimes require extra work.