So I want to sell my 100 shares of AAPL to him at a price of 10 or
even 1 US Dollar. Is that legal/allowed?
Of course. It's your stocks - do with it what you want.
if the two persons are not served by a same broker.
You'll have to talk to your broker about the technicalities of the transaction.
if the person who sell are US citizen and the person who buy are not, and and vice-versa
Since you asked specifically about US citizenship, I'll assume you're in the US or the transaction is taking place in the US.
Citizenship has nothing to do with it (except may be for economic sanctions against Russians or Iranians that may come into play).
What is important is the tax residency status. Such a transfer is essentially a gift, and if you're a US tax resident (which doesn't correlate to your immigration status necessarily) - you'll have to deal with the gift tax consequences on the discount value.
For example - you have 100 shares of AAPL which you sold to your friend for $1 each
when the fair market value (FMV) was $501. So essentially, the friend got $50,100 value for $100. I.e.: $50K gift. Since this amount is above the annual $14K exemption - you'll have to deal with the gift tax and file gift tax return.
There are also consequences for the capital gains tax for both you and your friend.
I suggest you talk to a licensed tax adviser (EA/CPA licensed in your State) about the specifics given your circumstances.
If you (or the recipient) are also a foreign citizen/tax resident - then that country's laws also may affect your situation.