Most reputable lenders (brokers/banks) and title companies will give you a copy of all the documents you sign at closing.  These documents are generally copied prior to you signing the actual closing package, so they won't have your signature on them.  However, they should serve any purpose you have in the future.

Anytime you close on a refinance, you should be provided with a complete copy of your closing package.  This allows you to carefully review all the documents you signed for your mortgage on your own during your three day recission period.

It's also a good idea to call your lender prior to closing and request a copy of the documents you will be signing at closing.  With today's automation, it shouldn't be a problem for most lenders to provide a copy prior to closing.

Remember, your HUD1 Settlement Statement is supposed to be available for your review 24 hours prior to your closing.

The attorney or title company should have given these to you at closing, but if they did not or you have misplaced them, you can try the following:

1. You should contact the closing attorney or title company used to close the transaction (probably easier than option #2 or 3). 
2. You can call the mortgage company that you obtained you loan from, if applicable.
3. If you did obtain a loan and it has been transferred to another lender/servicer, you may also ask them if they are able to provide this documentation (this is probably the most difficult option) .

FYI, depending on what you are looking for and where you live, many counties have certain items online (such as Deed of Trust/Mortgage)

I have never been to a signing where the escrow officer did not give the borrower a copy of the documents by either paper or disc.  If you have signed already and you do not have copies, you should be able to still get a copy of the documents from your escrow officer.  They are often emailed to them and they can print out a set. 

If it has been a while since you signed and the loan was implemented, then you can get some documents from the recorders office, such as the Deed of Trust and any additional riders.  For the other documents, you would need to contact the lender that has the loan.  They are required to keep certain documents, such as the Note, Truth-In-Lending, Good Faith Estimate, etc.  Best of luck to you.

You can bring a copy machine with you, or you can demand to have them copied as a condition of your signing them.  Many of the documents prepared today are done so with computers.  Frequently, the people using those computers do not know what they are doing, hence a great deal of mistakes.  Some mistakes are innocent, while some border on the criminal. 

The most important thing you can do is to stand your ground.  If they say thay 'can't' give you copies, say, "no; you can't don wax wings and fly around the room, but it appears you simply won't give me copies."  Nothing binds you legally until you actually sign the papers, so don't be bullied by statements like 'your are locked in', that is just silly and may be illegal.  Don't be afraid to walk away from the table.  These people are counting on the fact that you want the place, and will sign anything to get it. 

Above all, don't sign things you haven't read and don't understand.  The reason the morgage market has collapesed in this country is because people blindly agreed to mortgage terms that they had not read.  You would not sign a contract written in Russian [unless you were fluent], so don't sign a loan document that is functionally equivalent.