Great question. Coming from a studio perspective, I tend think of discrete roles for the Engineer and the Producer - even if they're they same person who switches hats depending on the day. The Engineer is typically the person who sets up the microphones / lines, get things set up for accurate recording, sets up temporary mixes for multitracking / overdubbing, etc. The Engineer essentially captures the ingredients. The Producer typically is the person who takes those raw ingredients and assembles them into a song. This means the Producer has a role in shaping the tones, adding effects, selecting takes, and even editing the form ("let's cut that third verse for the radio version"). The Producer may be making these changes when the band is not present in the studio. You put a lot of trust in that person, as they can help make or break your masterpiece.
All sound operators have a the basic responsibility of an Engineer. They should be able to take the inputs / mics and assemble a mix based on the relative volumes of each signal. Most reasonably competent sound operators are accustomed to having a Producer's role in the mix. They may decide the bass guitar is too bright and roll off some highs. That lead vocal would be smoother with some compression and plate reverb. Maybe the guitar would benefit from a bit of delay during the solo. Great sound operators are actively mixing the band during the entire show, acting as a +1 member of the band.
Like it or not, the FOH engineer is out of your control during the show. You can send them a signal and say "DON'T CHANGE A DAMNED THING IN MY SIGNAL", which is silly, and an insult to someone who is a good engineer. We want the FOH engineer to be the Producer, to take all the ingredients of the show and mix them together in the way that sounds best for the room. That is very subjective, and because the FOH operator will have their own taste about what YOUR music should sound like, they will have a large degree of impact on the product. If you want to help a Producer who is unfamiliar with your band, give them a clear reference for what you want to sound like, such as "for this band / song / set, we are going for a stripped-down raw sound, like that KISS album 'Dressed to Kill' ". If you don't give them an idea of what you want, you run the risk of a well-meaning FOH operator mixing you to sound like something completely different. Maybe they're really into the sound of Rush or Swedish Death Polka or Brazilian Technofolk. Just remember they will be mixing you while you're onstage, so you want them on your side.
With the AxeFX, I'm delivering a very complex signal to the front of house (FOH) operator. For those shows when we're using the house sound team, I always tell them I'm giving them a fully realized signal via 2 XLR lines, including effects, amps, cabinets, cabinet mics, and room ambience. I also tell them: "you're the FOH sound operator, so please start with a flat EQ on my channels, then do whatever you feel is appropriate to make the overall mix sound good, and after the show, please tell me if you have any suggestions about what I should change in my tone to give you a better / more usable signal". Even if they are skeptical about the AxeFX at first, the FOH operators always tell me after the show that they LOVE the AxeFX, and that it made their mix sound great. The typical feedback I get in terms of suggested changes is usually based on the room physics, such as "this room has a persistent standing wave delay coming off the back wall, and your slapback delay on that one song made the mix a little messy" or "I dialed down some of your low-mids because our subs are kinda muddy".
