I can a simple bluetooth module receive and transmit data at the same time? Or will it require a secondary antennta or even another bluetooth chip?
The term you are looking for is “full-duplex”.
“Bluetooth technology provides the effect of full duplex transmission through the use of a time-division duplex (TDD) scheme.”
[Edit: since it gives you the effect of full-duplex you don't need a second antenna/module.]