I would offer there's a third type of speaking partner: a paid tutor who is untrained (i.e. not a professional teacher).
A professional tutor will most likely have materials prepared ahead of time. Of course you can also do preparation, such as listing out questions you have, or bringing material you are working on. But usually not much lesson planning will be required on your part. If they are a skilled teacher, they will be confident in leading you through the lesson.
For the untrained types, you will need to put more effort into preparing the lesson. This can actually be a good thing sometimes, because it forces you to commit more to the learning process. This will happen a little differently with a language exchange vs. a paid untrained tutor:
For a language exchange partner, since you are really only
"paying" them by trading your language skills for theirs, there's not a lot of incentive for them to learn to be a good teacher. They're in it to be a learner, not a teacher. These types of relationships have less commitment, so most likely you'll end up cycling through a lot of exchange partners. The best thing you can do is stick to a language course, whether a formal class or self-learning, and use your partner as supplemental conversation practice. Try preparing a list of words you want to practice, and see if you can tell them a story. Have them help you say it right, and especially listen to pronunciation. These are important skills that are hard to get without talking to a native speaker.
If you have a little bit of extra funds to pay an untrained native speaker who will commit to meeting with you for a set time, you can really gain a lot from this. For this, you don't necessarily need another course. You can prepare the lessons yourself and train your teacher to teach you. There are several great methods for this, such as Total Physical Response or Growing Participator Approach. An example activity would be bring a set of pictures and have your partner point to each one and say it. Have them repeat multiple times. Then have them say a word and you point to the correct picture. Finally, have them point to a picture and you say the word. It will take more effort to get your tutor to lead the activities than a professional tutor, and it's always a risk because some people turn out to be natural teachers and some don't. But if you find a good one, it's a worthy investment.