I'm currently working on an F1 visa (in a few months I will move to an H-1B). For what it's worth, I pay federal, state and local taxes; but I don't pay social security and medicare. I'll start paying those two on the H-1B visa.
When I got my first paycheck, the company had deducted the social security and medicare. I went to the payroll department and talked with them, and they returned the amount a few days later (and haven't billed me for that stuff ever since).
I'm assuming J-1 stuff should be close to F-1.. But of course you never know
Good luck!
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