What should I look for in a personal monitoring system for a senior?

The device I think you're talking about (no monthly fee) is made by Assistive Technology Services (I think I saw it advertised on this site, in fact). Mom had a Lifeline, but couldn't remember why she should push the button or grasp the concept that she could not talk and hear with the button. She's also outdoors some and couldn't talk into the speaker base if minor assistance was needed (no 911 needed). So Lifeline went back and I bought this system. She (and I) love it! You have a choice to set up to call 911 only - Friends only - or Friends then 911. It is very easy for her to understand and will cover up to 600 yards from the base unit. Very easy to set up and very user-friendly. All she has to do is remember that when she needs me, push the button and she can talk to me. The unit can be moved from one residence to another and 'knows' to contact the local 911 wherever the unit is. It's a sequential dialing system - if the first person does not answer, it dials the second, etc. (Mom only pushes button one time for all to happen). To bypass answering services, "friends" hear a message about 10 seconds into the call telling them that it is an emergency call and they must press "5" to continue to talk with the person. therefore, if it rings an answering machine and the "5" is not, of course, pressed, it dials the next number. The people were helpful and the unit arrived soon after ordering.
If your mom drives or goes away from home, check out the "5-star responder" from Great Call. Mom couldn't grasp the responder unit button concept, but the feature is available on the jitterbug phone and is great (if you buy from someone besides Great Call, I've heard you have to ship the unit to them to set up the 5-star feature). Hope this helped.