You can operate as a tech in any US possessions such as PR or the US side of the VI, no problem. However for other countries the rules are different; you should familiarize yourself with them. I believe the ARRL has a site that lists each of the countries with which we have reciprocal op agreements and what the requirements are. I haven't done this in a number of years, and the rules have changed. I could, for example, use ham gear in Canada as a visitor by signing my call followed by the prefix of the province I was in.
 I found the ARRL website that talks about reciprocal op regulations...
As long as you're in the US Virgin Islands, there shouldn't be an issue. You will of course need to operate as KP2/KD8LUI in your case. In the Caymans, you're subject to the IARU reciprocal agreement, and must operate in conformance with Cayman Islands amateur radio law. Again, you will also need to sign as ZF/KD8LUI in your case.
As stated above, do your research on local laws before going QRV.
As long you're in a territory where communications laws are adminstered by the FCC (any US territory), you can operate as normal. Using W2MJR in his USVI example, you do not need to sign KP2/call when there, as it's a US territory. If you were to operate an HF dxpedition on the USVI, you would sign call/KP2 to indicate you were in the USVI to DXers.
You would, however need to sign ZF/call if you were to operate in the Caymans, though.
Again, research the requirements before going! Follow Mike's link.