Both a guitar amp and a stage monitor are essential actually.
Professional players will use their amp for their own sound, and that way the engineer can mic the amp instead of having the guitar feed the mixer directly (more sterile sound, but works good sometimes).
Then they will ask to have a mix of the rest of the band in their stage monitors. Some like also a bit of guitar reinforcement in their monitor, so they feel the sound surrounding them from 2 sources.
Regarding bass, it is less essential to have an amp, as most monitors can handle the load, and many times there's no need for high levels, as the bass frequencies can be heard on stage from the main loudspeakers. However, it's nice to have a good amp, for tone and again, seperation from the monitor mix.
When engineering FOH, I find that with musicians that have a good tone and can handle reasonable stage leves without cranking up the knobs, the amp can be next to the wall facing the crowd.I just add some on the FOH and no excessive fader riding is necessary. With less optimal settings, and my rule of thumb, is to have the amp on the side wall facing the bassist and the band. That way others hear the amp and less monitor distribution of the bass is necessary. I don't like it facing the stage, as the crowd is left with muddy bass sound (highs are lost) and it doesn't blend well with the FOH.