When looking for the right co-founder, the best place to start is by spending time with people. And not just a coffee meeting. Spend time with them for breakfast, lunch, dinner, and the bar. Figure out if you like the person in the morning as much as the evenings. Spend time talking about ideas, business philosophy, family, food, interests, life, etc. Just like when you date you want to find out if you have similar interests.
In your discussions, you want to find out three things.
1. What are they passionate about?
2. What kind of company do they want to build?
3. What is their cost structure and appetite for financial risk?
You need to be able to talk about your answers. It's one thing to write them down but you have to be able to talk about them openly. Talking about your feelings can be hard, but you have to. You have to ask questions that make both of you vulnerable. If you can't do this, you will never be able to have the type of communication you need down the road. A co-founder is like a spouse, you have to be able to talk about everything.
If you don't believe in the same things, end the dating. You CAN NOT compromise on what you believe in or you'll end up the cliche spouse in the relationship admitting you gave up on your dreams when you got married.
Along with similar beliefs you want to know they aren't going to give up. An easy solution is picking a fellow co-founder who has also started a company. Joining one early is nice, but finding someone who has been through hell and back as a founder is an invaluable experience.
If they haven't started a company you want to go back to their roots and understand what tragedies they have faced, what set-backs have they had to overcome, or what challenges have they beat. This can include a variety of life experiences from a rough childhood to losing someone to excelling at sports. You are looking for experiences they can talk about that took incredible personal commitment to overcome.
When you are in the dating phase with your potential new co-founder, BE OBSERVANT. Looking at small things can tell you a lot, such as how do they talk about other people, are they positive or negative, do they follow through, and/or are they on time? Small clues tell you a lot about the person, signs you want to pick up as early as possible.
Once you have established your new co-founder relationship I STRONGLY recommend that every founder (including you) earns their ownership over four years. For example, if someone leaves after two years then they get half of their equity. This is very, very important because you can't predict the future and you can't afford a massive amount of equity being held by a former co-founder who is no longer involved with the business. Relationships don't always work out and you have to structure the company with a fair way to reward people for the time they put in, while providing the company with an easy out if the person leaves fro whatever reason. Your company won't have cash to buy them out, so equity for time is the only way to go. If a co-founder is not willing to do this, walk away.
Getting to this point in the relationship is the most important part. If you have the same beliefs, a strong chemistry, totally open communication, and you know the person won't give up, you can figure out the rest. Determining roles, ownership structure, which idea to pursue, etc can be worked out. In fact, working these out together will better confirm your choice to take the co-founder plunge.