Can I transmit and receive at the same time? Does this vary by silicon vendors?
This questions requires a bit more knowledge of Bluetooth than just its existence, I am not entirely confident a simple and straightforward answer here will clarify enough.
Simply put: Yes, physically (that is, in the 'physical layer'), all BlueTooth standards are created as full-duplex. At least as is apparent to the layers above. (some details/caveats left out, because I am not writing a full white-paper here)
When using the low energy system, single mode devices may not be able to, especially when they implement a service that only receives or transmits. So, maybe, in your case, the answer is no.
But it still might be yes, if you are talking about a BlueTooth 4.0 / BLE enabled phone or PC, they most usually implement dual mode and full service stacks, allowing them to use full-duplex services, half-duplex services either in or out, etc etc (whether the software designer thought it was useful to allow your PC to act as the device in the heart-monitor service can be doubted of course - or in fact they thought to allow your PC to be a slave at all).
If you want to really understand how BlueTooth works and specifically BLE, you should really read up on a good source. I don't know any on-line "For Dummies"-like resource, so I can only point you to the BlueTooth organisation itself (no happy stories about bunnies and ponies, but all the raw info you need):
Even more specifically:
Each Bluetooth module has one name and can only do one. Tx or RX that is it. I use 2 modules 2 different manufacturers because of the name or same mac address.