1. We'll see each other next week. - This could be a spontaneous decision, a promise, or a certain prediction. The modal "will" is often used with "think, hope, suppose, guess, and imagine". It's often used with these adverbs: maybe, probably, likely, definitely, certainly.
2. We'll be seeing each other next week. - Using the future progressive for a planned or arranged action or event is similar, or the same as using the progressive form (taught as present progressive) to speak of a planned or arranged action or event. The future progressive draws attention to the action or event as one that is ongoing over a temporary time frame - or a limited amount of time.
3. We're seeing each other next week. - progressive - Use this when you are speaking of something that is planned or arranged.
4. We're going to see each other next week. - Use "be going to" to speak of something that is planned and decided. Be going to may also express one's strong intentions in a given context. So sometimes it's correct to use the (present) progressive or be going to.
Student or Learner