The privacy in a double close comes from the fact that they're two separate transactions with separate settlement statements. The seller sees the A statement, the buyer sees the C statement, and neither statement shows the other price. That's the mechanism, and it holds in most places.
It isn't absolute. Oklahoma requires profit disclosure even on a double close, so the privacy advantage doesn't exist there. Other states have added disclosure rules aimed at wholesaling, and some counties record deeds with the sale price stated on them, which means the A price becomes public anyway once it records. Roughly a dozen states don't make sale prices public at all, so what your buyer can look up afterward depends entirely on where the house is.
On the agent specifically: she has duties to her seller, and what those duties require in your state and under her brokerage's own rules is a question for someone licensed there. She may be describing a real obligation, or she may be describing her firm's internal policy, or she may be trying to get her seller a better price. You won't resolve that by arguing with her.
The practical thing is that the deal itself doesn't depend on the secret holding. If the seller learning you're reselling at 235k would blow up the contract, that's a signal the A price is one the seller would rather renegotiate, and you should decide now what you'd do if they ask to. Get whatever the agent tells you in writing, and take it to a real estate attorney in your state before you respond to her.
