I personally think it depends a lot on the child, the piggy, and the degree of parental support.
For example when our piggies were new (and very skittish) the children were never allowed to pick them up - only hold them in their laps while sitting down, and always with an adult next to them to supervise.
These days (2 years later) both girls regularly handle the piggies alone.
We know their personalities well, so for example DD1 is the only person who ever handles Ruby because she doesn't like anyone else and will bite.
Lucy is difficult to catch, but the calmest and most snuggly of all our piggies.
Eddi is just laid back and easy to catch and handle and loves a good snuggle.
And Oreo is easy to catch and does love to be held but is quite fidgety and squeaks constantly, so again it is mostly DD1 who handles her.
The exception to this is that in Summer when we are taking them outside for lawn time I always take them (even Ruby) because the risk of dropping one in the garden it too great to take.