I've done it several ways: I have a cargo net mesh that I use occasionally, and yes you can strap stuff to the passenger seat that way, but most of the time for grocery (or wine store) runs I just use a backpack (unless I need a lot of stuff I found it was the most practical).
And then to go to the office (or bigger grocery runs) I recently bought a Kriega dry pack US-30, because my office laptop is super heavy and it's not comfortable to wear a backpack with that thing in it (plus the backpack is not waterproof). So the Kriega makes it super easy to strap all this stuff securely to the back seat, I love it. It's just a little unwieldy to operate (lots of straps/buckles) and relatively big, so for small grocery runs it's not really worth it...
But so far (it's been over three years) I've been doing most of my grocery runs with the CB and it's absolutely fine! Love that thing, the only downside is that it's a little underpowered...
Amazon
Powertye 50159 Orange 15"X15" Cargo Net Featuring 6 Adjustable Hooks & Tight 2"x2" Mesh https://www.amazon.com/dp/B00T6JKH2C/ref=cm_sw_r_apan_glc_i_GWHJ36BKCVCSVHDJZKRT?psc=1
its this one
https://www.amazon.com/gp/product/B00T6JKH2C/ref=ppx_yo_dt_b_asin_title_o01_s01?ie=UTF8&psc=1
it works well for the large basket. I think the company makes them for the smaller ones as well.