Egeanin comes from a military (naval), not trading, family. She never mentionned her father, always her mother who was a Hand of the Empress at Sea. There's no reason to believe she's been to Shara. What we know of her youth is that she entered the Seanchan navy very young and rose through the ranks, shining enough to be chosen for the Return.

I hardly imagine Egwene warming up to Egeanin nearly enough to chat with her about her childhood anyway.

Egwene viscerally hates the Seanchan (they're her blindspot like Aes Sedai are Tuon's, she can barely even consider they're not all bad), she'll be even more reserved and cool than Nynaeve toward Egeanin, not to mention that Egeanin herself is not sociable and has never been one to share anecdotes, not even with Bayle Domon.

Not that Egwene necessarily will spend much time with Egeanin. Egeanin's there to provide Egwene with what she needs to hatch a scheme that will destroy the sul'dam/damane system and Tuon's rule in a matter of weeks or so. Egeanin's return to the south could even be necessary for the plans to work.