I don't think it's much of an issue.
As Radhil mentions in his comment, she's a mistborn, part of a noble house, Elend's ex (essentially), and trying to take down House Venture. I don't necessarily come to Radhil's conclusion that those allowed her to know what Elend was doing, but it at least would have gotten her in the ballpark.
But how could Shan know he was reading forbidden books in the first place?
Surely, she wouldn't have sent anyone to search for Elend's stupid books, had she not known what exactly she wanted to find.
I disagree that she wouldn't have searched if she didn't know what she would find. She would have known quite a bit about Elend's general personality, and attitudes towards the status quo, from their involvement together and just general gossip between noble houses.
As a high ranking noble and mistborn, she is expected to have a pulse on the noble community, and have a good idea what noble houses are planning and what the high ranking nobles are doing. She likely knows about Elend's meetings, and that something is going on. Sending her servant over to scan through his books is just another aspect of the espionage that the noble houses employ against each other.
Servants, and even the nobles themselves (especially allomantic ones) likely spy on each other on a regular basis. Being nosy about the books someone is reading is just another prong in the gathering of information, which is required if you want to try to move up the noble ranks, or even keep a high rank.
It makes perfect sense to me that Shan would send her servant over to spy even if she didn't know the contents of the books. I feel like if she knew, she wouldn't have bothered bringing the servant to check, and would have instead used that information to get Elend/House Venture in hot water.