There are a lot of ways to think about your martial art - is it a form of physical culture, a means of learning self defense, a cultural/ historical preservation activity, etc. None of these are wrong, and in many ways they aren't exclusive - you can train to get in shape AND to preserve a part of history that you find valuable, for example. The way you think of your art will, however, shape your training in many ways (if you train to preserve a cultural artifact you're less likely to incorporate techniques from other arts, for example), as well as shaping your marketing (sales pitch) and the way you may structure activities around your dojo.
I've already written about how I think we can benefit from regarding our training from a physical culture perspective (and I stand by what I wrote there). I have another perspective to share today.
I tend to do the bulk of my training alone. I go to class when I can, which is often just once a week, but I'll practice kata and kihon or do my conditioning by myself several other times a week (by the way, this is probably a bad idea for any beginner, but I've got many years of supervised training under my belt). There are advantages to this approach. I have less travel time. I can train whenever I want - I'm never late for class! I can focus on the skills I need most and on the exercises that work best for me.
There are also some pretty obvious disadvantages. You can't do partner drills by yourself. It's very difficult to work on many key skills without a partner - you can get clean technique, but how do you develop your timing? It's even harder in grappling arts, I'm sure - how do you even practice throws, holds, and locks without another body to toss around?
I think that many people will also suffer from a motivation gap. I'm not that way personally, but for many people it's easier to show up to a class and train because someone is telling you to train, rather than trying to force yourself to practice alone in an empty room when you could just go have a beer instead and nobody would be any the wiser.
Let's put aside these considerations for now - these points all have to do with the fact that training with other people can be better for your karate. They're valid points, but there is another set of advantages to training with a group that have nothing to do with improving your skill:
The thing is, humans are social animals. We evolved to travel in groups and there are serious biological implications to that fact. There is a substantial body of medical literature showing that human health is enhanced by having strong social connections - friends, family, whatever. We can argue over some of the fine points - do internet friendships count or not (I suspect they do) - but the bottom line is, very clearly, that people are healthier and happier when they have a deep social environment.
You can get friends from a lot of places, but one very interesting social bond is created by shared physical suffering. Anyone who's ever been through a football camp or a hard promotion test can tell you this. In my dojo we often hug or pat each other on the back after tough sparring matches. This is why corporations spend tons of money to take their executives rock climbing or white water rafting and call it "team building." It really does build a sense of togetherness and bonding among people.
The people in your dojo a) share your interests; b) spend time with you regularly; c) share bonding experiences regularly. They're also unlikely to be total douchebags, at least in a good school, because if you have good seniors the douchebags get their asses kicked hard enough that they either straighten up their act or quit before getting very high in rank (usually).
My point? You should be (and probably already are) friends with your senpai and kohai. You should hang out with them outside of class. You should invite them to your Christmas party. You should share birthdays. You should be Facebook friends. You should hang around before or after class and catch up with them.
Furthermore, being friends with your classmates is a key benefit of your training. You might justify watching a football game with your buddies or poker night by saying that you need time with your friends - the same is true, perhaps even more so, about training. If you're thinking about spending money on a martial arts retreat or attending a seminar or having a social function, and you're waffling, don't think of it in terms of just how much karate you'll learn - think of the benefits of that activity as a social occasion. Karate can help fulfill your fundamental human need for interaction!
If you can (this will depend on the culture at your school), organize and participate in social activities outside of class with your peers. Do a martial arts movie night, have parties, go for drinks after class, whatever. And don't resist these activities because they might have limited direct benefit for your karate - that's not the point. The point is to enhance your health and happiness by deepening positive social connections.
If you run a school, encourage your students to socialize outside of class. Get them to do movie nights (or organize them yourself). Be available to grab drinks or snacks after class, at least some of the time. Put up a board in your entrance area for people to advertise get-togethers. You might be teaching self defense, but you're also creating a social network for your students.
Want to convince someone to start training? Yes, they'll learn to defend themselves, and yes, they'll get in better shape. But they might also make a whole bunch of new friends, and that has an added value all its own - not just int he obvious ways, but enhancing health and longevity.
If you think you can be fit and healthy and a loner... you're probably wrong. You can get a social life by going out drinking and partying multiple times per week, but you're probably going to last longer if you hang out with your dojo mates instead!