The requirement that a creditor must verify debt after receiving a timely dispute from the debtor is based in the Federal Fair Debt Collection Practices Act (FDCPA). This requirement, however, is quite broad and does NOT require that the collection agency provide you with a copy of the contract. It essentially requires them to look into your account and verify with you that indeed that balance is due and owing. In most cases this means they will send you a billing statement of some kind that shows the balance. In many cases a debtor may not be satisfied with the information sent to them to "verify" the debt.
If you have a legitimate dispute to this debt, or believe the balance is wrong, and the same debt keeps coming back to you through different collection agencies, then consider seeking a consultation with a consumer law attorney who can assess whether there are any improprieties on the way the debt is being collected.
Also, remember to keep copies of any letters you send out, and keep good records of phone communications.
They should be sending you the contract and some evidence that it has been assigned to them. Your experience is not uncommon. Many collectors only have an electronic record that is insufficient to validate the debt, so they just go away. Some may resell the debt, and you may be approached again in the future. Just keep doing what you are doing, it seems to be working!
You will typically only get recent statements from the underlying creditor or a copy of their debt collection notices, pre-turnover. These, of course, don't verify much, if anything. I also encourage clients to check their credit reports post FDCPA dispute/validation requests sent to the debt collector. If the debt collector is reporting but fails to report the debt as disputed, you now have Fair Credit Reporting (FCRA) rights (and hence leverage) to settle the debt and eliminate the credit blemish. Good luck.
This answer should not be construed to establish an attorney-client relationship. Please consult a licensed attorney in your state and visit www.FloridaBKLawyer.com or www.ConsumerRightsGroup.com for more details.