I am a non-US + non-European citizen who is doing a postdoc in the US (under the J1 visa), in parallel, I have a full-time job (remote) in Germany. What will happen to my tax, shall I pay any extra taxes rather than what is being deducted from my payrolls?
In the US, while you may not have to pay any extra taxes, unfortunately you will have to file a tax return. It is not automatic.
Start with the assumption that you are filing as a US resident (typical for J1 status). Use tax software to work through the questions about foreign income. Even if you have to declare the German income in the US, you won't necessarily be double-taxed, because the US and Germany have a tax treaty.