If you were waiting for a native speaker, I'm here.Don’t give up on them altogether. = Don't lose hope/faith in them/their ability to accomplish something.
Don’t give them up altogether. = Don't turn them over to someone else./Don't abrogate responsibility for them./Don't disown them.
Hi dn88,Hello AWordLover,
I do agree with you both, but here's an example I have invented to show that "give sb up" and "give up on sb" can be the same.
You've been waiting for your friend for hours and you've nearly lost your faith in his arrival. Finally, he turns up to your great surprise. So you say to him:
I've been waiting here for hours - I'd almost given you up. [A native speaker would certainly know what you meant, but may think that you have made a mistake. Give you up, would be an incorrect form of give up on you.]
I've been waiting here for hours - I'd almost given up on you.
Am I wrong? Do the two variants above convey the same idea? I think that either of them means: I had almost stopped expecting that you would arrive. If not, please correct me.