My grandma says children should never be looked after by nannies or baby-sitters. Do you agree with her?

Personally - I prefer that relatives watch my children if I can't.  So, in a way, I do agree.  A family member is better than a stranger.  

In modern society with expenses so high, it can sometimes be unrealistic to think that one parent would be able to be a "stay at home" mom or dad.  And, at some point, parents do want a night out and a short break from the kids even if it is only for a few hours.  For this reason, it is not realistic to think that children would "NEVER" be left with a babysitter or a nanny.  

At some point, children are left with someone else - grandparents - aunts or uncles - older cousins - someone else will end up watching the kids at some point.  Anyone who says their children were NEVER left with someone else is probably lying.  

No I don't agree with her. 

If they're trust worthy. I'd say that spending time with your own children to create a bond and be a positive parent and show them right from wrong is a very important role as a parent.