You don't absolutely need to install a new O2 sensor, however there are some huge drawbacks to doing so.
As the mechanic stated, it doesn't affect how the engine runs. The downstream O2 sensor (in your vehicle) only checks the efficacy of the converter. The drawbacks to running without the sensor are, your Check Engine Light (CEL ... what you called "engineer light") will always be on because it cannot actually check the cat ... there's nothing there to check it with. This could potentially mask some other problem which could pop up. Due to this reason alone, I'd suggest putting a new O2 sensor in.
There may be an issue when installing the new O2 sensor, that being, there may not be anything to plug it into. A thief is going to do their job as quickly as they possibly can. If they used an angle grinder to take it out of the exhaust system, more than likely they also clipped the wires going to the sensor in the same manner. You might be lucky and they pulled it the "normal" way, but you're going to have to verify it before you can install the O2, then if it isn't there, procure the new pig tail and install it.
As far as O2 sensors go, OEM is just fine. Walker should be okay. Denso is another one which would work fine. Buy from a reputable source and you should be golden. While we don't really do recommendations around here, if I were to recommend an aftermarket one, it would be Denso for the only reason they provide a small tube of copper anti-seize which works great for removing the sensor the next time you have to replace it (hopefully not for the same reason).