As per the title, but I mean in theory, not specifically Sentinel-1. Is a bistatic antenna required to get all four polarisation modes?
Monostatic means that the sending and receiving antennas are the same, i.e. one instrument. Monostatic systems can be fully polarimetric.
One example of monostatic SAR system is Radarsat-2 for some of its modes, for instance the standard Quad Polarization mode.
Standard Quad Polarization
In the Quad Polarization Beam Mode, the radar transmits pulses alternately in
horizontal (H) and vertical (V) polarizations, and receives the return signals from
each pulse in both H and V polarizations separately but simultaneously. This beam
mode therefore enables full polarimetric (HH+VV+HV+VH) image products to be
generated. The Standard Quad Polarization Beam Mode operates with the same
pulse bandwidths as the Standard Beam Mode. Products with swath widths of
approximately 25 km can be obtained covering any area within the region from an
incidence angle of 18 degrees to at least 49 degrees.
Other SAR sensors have this capacity as well, but not Sentinel-1