Writing this from one of those destinations, where I have been for a week.
The answer is fairly simple. A flight from Europe to the Carribbean/Mexico is in the order of 9-12 hours outbound and 8-10 hours inbound for the most part. Given a two hour turnaround and a 90 minute report, the average rotation start to finish is 21-25 hours. Obviously the rotation requires two full sets of crews. On top of this there are considerations for minimum crew rest and allowable duty periods that are in turn influenced by the fact a crew may or may not be accimatised to the local time zone. It takes 3 local nights free of duty to become acclimatised to a local time zone in this particular case.
A crew of 10 to 14 people requiring a one way airline ticket is also a very considerable expense, that often far outweighs the costs of allowances and hotel accomodation at the resort location. Multiply these one way costs by a factor of two and allow for the fact that a crew still needs to be pre-positioned at a destination to operate the return flight with adequate pre-flight rest to satisfy the normal requirements, as well as any contingencies that might arise, (cancelled or delayed positioning flights being but one example) and the economics of leaving a crew abroad for a week become very attractive. In addition by the time you have otherwise positioned a crew home and provided the requisite rest periods there is in reality, little productivity gain in any event.
If a company operates more frequently than once a week, then obviously the previous outbound crew will likely operate the next cycles inbound flight. It is also possible for crews to be used in a similar fashion within a geographical area where positioning availibility is a practical proposition, bearing in mind the rest requirements and contingency planning already mentioned.
So the short answer to your question is yes. Crews often do remain at the destination for a week.