Let's start with pointing out that mathematics and formal logic are two things that do not apply to natural language in the sense that one can not follow clear rules to simply translate a statement in a natural language into some logical construct following simple rules. One has to understand the semantics of the natural language, and often, context is everything.
A simple example is the question would you like coffee or tea? which looks to a mathematician like a simple yes/no-question. Try answering yes to your colleague next time you are asked this question...
That being said, let's see what your sentence translates to:
Neither L nor S lives in either A or B
The first part Neither L nor S means that whatever follows does not apply to L and does not apply to S:
NOT (S OR L) = (NOT S) AND (NOT L)
The second part in either A or B means that whatever went before applies equally to A and B.
(P is TRUE for A) AND (P is TRUE for B)
So the proposition that X lives in Y, applied to this sentence, means:
((NOT S) lives in A) AND ((NOT L) lives in A)
((NOT S) lives in B) AND ((NOT L) lives in B)
So your friend is right: S does not live in A, S does not live in B, L does not live in A and L does not live in B.
What I think is "One(L or S) cannot live in the places that has been mentioned(i,e A or B) which makes the other one to live in A or B"
Nothing in the given sentence justifies such a conclusion. If I tell you that Alice does not live in New York, how would that possibly imply that Bob does live in New York? Nobody ever mentioned that either L or S should live in A or B.
It looks like you fell for a false dichotomy. The fact that one thing is not true does not make an unrelated thing true.