How to determine the FFT size of LTE downlink receiver? I'm puzzled about the LTE FFT size. 

Assuming an LTE base station works at 20M mode, the FFT size is 2048. Then, many users can be served by this base station. 

Do all these users work at 20M mode (i.e., the FFT size of all these users are also 2048)?

OR

This 20M base station can serve users working at various modes (e.g., 1.4M, 5M, 10M, 15M, 20M modes)? For example, one user works at 1.4M mode with 128-point FFT and one user works at 10M mode with 512-point FFT can be served simultaneously by this 20M mode base station?
You don't need to sample at cell FFT rate. Just shifted the received signal to the desired subcarrier and take a lowpass filtering, then sample at user-specific bandwidth (see attached image). The shifting operation can be implemented by a multiplication with phase ramp in the time domain.



However samping at cell specific FFT rate is optimal because the lowpass filter needs some kind of roll-off.