I took her use of words "logged on" as meaning, she went onto her mums fb profile and went thru her friends list (which are visable) and contacted everyone via PM, hence why many were contacted next day, these friends may not have received her message straight away if they are not frequent fb users, or if Sarahs message went into the approve request area first (which is easy to miss). Otherwise to log onto her mums page she would have to have Karens password.
Also if she had "logged on" to Karens page, I would have thought she would put a great big message along the lines of....... hey everyone this is Sarah, Karens daughter, mum is misssing and we are very worried. Please if you have heard from her or know where she is get in contact with me on (mobile no.) or contact police.
She could have posted a message on Karens wall, if it's only made available to friends we wouldn't know. Also possible that she sent each a msg from Karens account to ensure that they have been seen.
I think she does know or has guessed Karens password, and has logged onto her account.