There are different ways, and the personal preference of the bassist / of their sound person matters. If they like their current amp (for their sound qualities), a lot of sound engineers will just put a microphone in front of it to pickup "their" acoustic sound ; if the amp itself is decent, it's going to feature a line output on the amp itself, that they can use to plug into the sound system and get the amp feed up to it. And, lastly, inserting a DI box between the instrument and the amp allows for a split output of one line towards the sound system and another towards the amp, so they can still hear "their" sound while the sound engineer has full control of how it will sound on the system (DI's are usually provided by the stage, rarely by the musician, and using them is a lot of time the choice of the engineer).
In a cool setup, sound engineers like to have both a microphone in front of the cabinet and from the amp output or a DI, allowing them to mix between both.
In a modern (bigger) system, the amp is entirely removed, replaced by a DI, or the cabinet is out backstage to avoid noise pollution on stage, and the musicians only hear themselves through the monitors.
Am I answering your question? No I'm not!
Talk to your friend. They may like their current amp to the death, they may very well know what they would like to buy next, or not.
Why not finding a pretext to go to a music shop and try several "just to have fun / just to have a look"? If an obvious preference arises, you just pop your credit card and Bob's their uncle!