I don't know that she'd want to announce her correspondence, but I also don't know how easy it would be to conceal it. Unless she asked the servants to give her her post separately and not mention it to anyone, the family would know that she's maintaining a correspondence with SOMEONE. Maybe she could just sort of imply she's writing to Charlotte more often than she does? I don't think she'd lie about it. She'd also have to make sure Catherine didn't write a return address on the outside of the letter. But what about Elizabeth's own letters? She can't just leave them in a pile with the rest of the post to go out, not if she doesn't want people to see the address; she'd have to either enlist the servants or take it to the post office herself. Which means she'd be paying the postage, which means it would be coming out of her pin money. I don't think she can entirely conceal it even if she wants to.

I think Mrs. Bennet and Lydia are unlikely to notice unless someone else brings it up. I don't know if there's any way to defuse the Lydia bomb, but Mrs. Bennet might start angling for Elizabeth to be invited to stay with Miss Darcy (hopefully along with one of her sisters!) because there are sure to be rich men in Miss Darcy's circle.