Is it possible to receive signals properly from the transmitter within a short range without a low noise amplifier in the receiver, or does the LNA provide more than power amplification, like helping block unwanted signals?
LNAs first task is, of course, Low Noise Amplifying.
Whether it is needed or not should be made clear from a link budget analysis, losses, gains and margins are summed up and compared against available TX power and RX sensitivity for the required S/N or BER.
The second point you've arisen is filtering, that's often included in the LNA block but is however functionally separated, it could live on its own.
In other words if you only need filtering, just include a lumped elements filter, a SAW or some microstrip so to match your requirements.
Finally another important feature of LNAs is reverse isolation, they can help avoiding local oscillator or digital noise from your circuit being routed back to the antenna system and radiated.
This will help with EMC compliance