If a couple wishes an annulment for a marriage that was conducted in an SSPX chapel, the prior of the chapel conducts an investigation and sends the docuмents to Menzingen. Menzingen then decides if the case is strong enough and then forwards the docuмents to "Rome." Rome treats the request as if the couple were married in Rome. Roman authorities grant the declaration for nullity. I believe the Latin American province has dealt with this a lot.
In a similar way, if a priest wishes to return to the lay state (due to previously unknown impediments, etc.), previously, his file would be sent to Menzingen which would then conduct a preliminary hearing and forward the file to Rome. Now, Rome has granted the authority for this particular task to Bishop Fellay because they believe that this would constitute an "internal matter" which could be handeled by SSPX authorities. The reason why marriages can't be annulled by Menzingen is because (similar to confession), the sacraments are "juridical sacremental acts" where "juridical" authority comes from Peter down to the Ordinary down to the priest. It's a little complicated.
Archbishop Lefevbre and Bishop Fellay have cited the above (^^) as one of the many reasons why it is important to keep a constant line of communication with Rome despite all the problems.
On a similar note, this same "privilege" is totally absent by other traditionalist groups because Rome does not recognize the validity of priests ordained by Thuc line bishops (after a certain time period) and, as a result, consider the marriages as equal to protestant marriages (as in, absent of form/matter of the sacrament = no sacramental ever took place). It's extensively docuмented that many "traditional" priests from the Thuc line that have left their religious communities, have appealed to Rome to be returned to the lay state. I know of a few cases, just from the top of my head, where Rome simply stated that, after an investigation done decades ago by Cardinals Ratzinger and Bertone, there is enough doubt to consider sacraments performed by such priests null.
I hope this ^^ helps with some of the questions/comments that were brought up.