They are normally pets to watch and not touch.
I have a society finch who is 7.5 who trusts me, will come to me to see what I'm up to and eat from my hand. It's a bond forged through a long time of letting him out in the house and letting him choose to interact without being pushed. He doesn't step up on my finger or command or allow any touching, and mostly does his own thing with his mate. He's just not scared of me anymore. Most finches will be, especially in the beginning.
There are exceptions to the hands off rule. Finches raised on formula from chicks, especially Java finches, never learn any fear of you and are bold and curious. These birds can make excellent pets. I hand raise javas, and the difference between my babies i raise and their wild parents is vast. The hand fed birds jump on you and play like puppies!