Can you be a "moral" person without a religious base for that morality?
If you can base your morals on the religion of your choice you can base it on whatever idea of your choice, quite fine.
If so, what do you base your measure of "right or wrong" on?
Pretty much on what everyone hopefully base their morals on: A mixture of considering the well-being of ourselves and our common beings, of empathy, of our idea what makes this world a more enjoyable place and what´s required to get there, and of a certain amount of common sense.
How do you judge others for being "immoral"...
The primary goal of having moral stance isn´t - where I come from - to be able to judge others. It´s more like a means to guide myself.
say a serial killer who actually "gets off to killing people" (is missing the part of his psyche that allows him to feel shame or guilt) ....
Why would I have to judge him? I just don´t want people to be killed.
I´m glad, though, that you picked a moral stance you and I agree on.
A much harder question would be: What - other than the appeal to a God who commands people to be serial killers - would allow to hold the moral view that serial killing is
right/good?
if he doesn't think what he is doing is wrong, is it really immoral if we allows individuals to define personal moral standards?
Well, people have their ideas of what are good things to do and what aren´t, no matter whom or what (alleged) source they may ascribe these ideas to. People can´t be kept from having their ideas and people can´t be kept from believing whatever source they believe them to have. So I am not quite sure I understand what you mean by "allowing them..." or what "disallowing/prohibiting such ideas" would mean, practically.
What, more importantly, I don´t seem to understand at all: How you arrived at the conclusion that claiming "this is what the God of my concept says" exempts you from being just another person who has his individual idea of morality. I.e. the relevance of religion for the question you asked.