Is it a negative thing, a positive thing?
In moderation, I think it's fine. Too much and I think it can be very unhealthy.
Is it an indication of faltering self esteem or insecurity?
Some insecurity - sure. Faltering self-esteem - only if excessive.
Does it show they truly care about you?
It shows they care about losing something which is not the same as caring about you. It might show they care.
Does it show that they don't trust you to be faithful?
I don't think it has to - no.
What does jealousy in a partner say to you?
I just think some people have more leanings in this regard than others and it probably varies at different stages in a relationship.