To say 'there, there' do I have to touch someone's shoulder?
If so can I say it while touching another part of the body?
There, there, don't cry. Everything will be OK.
Or can I say it without doing it?
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"There-there", as noted in a comment, does not really mean anything. It is an expression used when comforting something. Just like saying "Bless you" when someone sneezes does not really mean a blessing.
"There there" is well described as a "comforting verbal gesture". It goes naturally with other comforting gestures, such as gentle contact on a non-threatening part of the body, such as a shoulder. Depending on your relationship "gentle non-threatening contact" could be a hug, a pat on the head or a back-rub. But you can say "there there" without touching.
I'm not sure how common it actually is. The context is rather unusual, it would normally be a parent comforting a child (and so not a context that English Language Learners are likely to be in, though it might be used for a piece of fiction) In this context you would probably have picked up the child and be making lots of hushing and soothing sounds, but I suspect this is common across languages and cultures. I suspect it is rather unlikely used adult-to-adult, but this kind of language is poorly recorded in corpora of written English.