80% of human communication is nonverbal.
Autism, among other things, means a difficulty to some degree in reading nonverbal communication. Let's say you're mildly affected and "get" between 60 and 80% of the nonverbal communication an NT would. You're still missing a lot.
Autism, among other things, means not feeling the same things, at the same time, in the same way as "normal" people. This results in a lot of getting kicked and called names, even in a supportive home environment, before your peers learn how to "act human." Not to mention from adults who never quite make it all the way through puberty.
Missing links in communication (and you know they're there)+miscommunication+getting kicked a lot=constant or near-constant worry about things normal people don't have to think about nearly as often (aka paranoia).
Yes, it is normal for Aspies to be paranoid.
That does not absolve us of the responsibility to take ownership of our paranoia, put a leash on it,and have it instead of it having us. It does not obligate our care-givers, friends, relatives, and loved ones to cater to our paranoia. It obligates us to learn to freakin' cope with it.
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"Alas, our dried voices when we whisper together are quiet and meaningless, as wind in dry grass, or rats' feet over broken glass in our dry cellar." --TS Eliot, "The Hollow Men"