At our school we don't bow to a picture of some mystical figure head. We also don't bow coming through the doors or to get onto the mats. These are traditions steeped in daoism and shintoism and we, not being daoists or shintoists, do not feel that the habit is necessary or relevant (or appropriate) to our entire school's population. We do however bow to each other. So why?
If we relate jiujitsu momentarily to judo we can find one very good reason. Judo is a sport sanctioned by Sport Canada. Clubs are privy to gaming grants and funding, and all coaches must be certified through the National Coaching Certification Program before they can teach minors or referee. Judo, like jiujitsu and wrestling and other grappling arts is a very tactile sport, requiring adjustments to posture, placement of the hands, feet, and any other part of the body. In order for a coach to do this in other sports, the coach must first ask the athlete for permission: "may I adjust your arm for a second?" In Judo, the permission is given when students bow in. The bow signifies consent. Judo is different from most other sanctioned sports in this manner.
Most schools have waivers which highlight that there are risks in participating in jiujitsu, but the waivers do not lay out what our behavior should look like. If we are assuming that bowing in is giving some kind of consent to our coach and training partners, it is important for us to know what we are consenting to. Besides the ability for a coach to make adjustments to technique, we are consenting that we are participating in a risky activity. It is important to understand that we have not signed a death waiver that entitles one party to slay another party at will without impunity. We are consenting that the inherent nature of the activity is risky. We are giving consent exposing ourselves to the natural risks of jiujitsu. What we are not giving consent to is overtly dangerous activities that fall outside of the normal scope of people who have day jobs practicing something they can leave and go to work the next day.
Jiujitsu is not without its frustrations. Often the frustrations stem from some part of the ego that refuses to concede, likely a survival mechanism. But in jiujitsu, this survival mechanism can be dangerous. This scenario often plays out in new students, especially where the new student is the stronger and larger and more often than not, younger of the two partners. This is when there is great risk of slamming, and explosive, uncontrolled movements that can be accidentally disastrous. This behavior is not within the scope of activities that we give consent to.
On the other hand, sometimes rolling can ramp up. Especially before a tournament when stress and anticipation levels are high. It is important to distinguish fighting like you are in a competition, and rolling casually with your training partners. Competition comes with its own set of unique risks that also comes with its own unique consent.
Jiujitsu is not as strictly regulated (as a result is not privy to the grants and funding that judo is), but it does share a trait with judo. It is a tactile discipline that arguably cannot be practiced without another person. As a result, training partners become valuable. The better suited a training partner is, the better your jiujitsu becomes. Indeed, the better the entire school becomes. So it is imperative to each students' progression in the art that our training partners stay healthy and uninjured. We had a great wrestling coach come to our school and he always said, "don't break your toys!"
With respect to bowing in, we observe the golden rule: Do to others as you would have others do to you.