1) It’s been ten years since we’ve seen each other.
2) It’s been ten years since we saw each other.
Are the above two sentences grammatically correct? If they are, what’s the difference between them?
I have always thought that we should use simple past after since in sentences like these. But I have seen a native speaker use the first sentence which put me at a little unease.
Can someone please clarify this for me?
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Both are in use; only #2 is accepted by careful grammarians like you.
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Nitpicking for good 
Which one is correct (or natural)?
But I have seen a native speaker use the first sentence which put me at a little unease.
But I have seen a native speaker using the first sentence which put me at a little unease.
Also, should we use singular (a native speaker) or plural (native speakers) for generalized sentences like the one given above?
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Which one is correct (or natural)?– Neither. You need a comma after ‘sentence’, and ‘put at unease’ is not idiomatic. I suggest ‘made me uneasy’. ‘Use’ and ‘using’ are both correct and natural.
Also, should we use singular (a native speaker) or plural (native speakers) for generalized sentences like the one given above?-- Either is fine. Native speakers more often use ‘native speakers’.
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put me at unease – I think I have seen people use this expression before. May be they are wrong too.
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Probably. Ms Google says:
211 for “put me at unease”
224,000 for “put me at ease”.
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This probably means that people are more often put at ease than unease. Good for them 
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That is not a likely conclusion at all, Daemon. It is much more likely to be proof of my point-- that it is not a generally accepted phrase with ‘unease’.
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You are probably right, MM. I was just kidding.