Anytime I sign a document, Do I have the right to a copy of the document I sign??

Yes. Only exception I can think of is if you are signing as an agent for your employer or something like that, in which case they would have a right to a copy of the document.

It's a matter of common courtesy to make a copy for you. There are, of course, possible circumstances under which a copier is not immediately available.

If necessary, tell the person that you will sign only if they promise to give you a copy.

Whether you are required by law to receive a copy of a document you signed depends on the type of document and the laws in the state where you are signing. For example, you are required by federal law to be given copies of all documents you sign when taking out a mortgage loan. But under NJ law a landlord is not required to give you a copy of a lease that you have signed.

Any time that you have to sign something you should make it a condition that you receive a copy of the document immediately after you've signed it.