Even though the buyer has their own lender, the seller/agent may require this process to take place.
However, something you need to know it may never have the clause "Must use the seller/agents lender to purchase the property in question." - this would be a violation of ethics within the real estate channels.
A typical request in text form in the MLS is, "Buyer must prequalify with _____________ in order to have their offer submitted to the seller."
Some agents get confused and tell their buyers this is something that does not have to take place. Indeed, that maybe true, it does not have to take place. In fact, if you don't your offer won't make it to the next step and that is being seen by the seller.
It's tough to fight city hall on this one. That is because, any real estate agent that has been in the business for at least two minutes will tell you that some lenders don't write their approval letters on paper that is worth anything.
Although, most of those "ships have sailed", there are some that will not look at any paperwork, such as VOE - Verification of Employment, Tax Returns, Profit and loss statements (self employed), or paychecks stubs, they will just write the approval letter "because of their relationship" with the agent representing the buyer.
While this may serve the buyers agent, it creates havoc between the other parties involved in the transaction. Including the buyer and their bogus letter, plus their Realtor.
While intentions are good in most regards, we still don't know if this "paperless" buyer will ultimately qualify for the home that they have written the offer on.
Without this cross qualification process taking place, the property could miss the offer that has a truly "qualified buyer" because of it being held up in escrow and off of the market.
Some of the more savvy realtors/sellers will say the buyer does not have to contact the sellers/realtors lender directly to give up all of their personal info. They are able to have their lender speak with the sellers/realtors lender, which will serve the same purpose.
Bottom Line, don't lose sleep with regard to having to cross qualify with another lender. If your lender is worth their salt, there will be no issue.
In fact, your credit does not have to even be run again. So don't worry about your FICO score suffering. Besides, if it is a rapid rescore will fix it. Be safe - Search well and let me know when you want me to represent you when it comes to real estate.