It could be set up many ways, depending on the specific arrangements. Either way, a HACCP plan needs to be developed for your product. You might have your own HACCP plan to cover your steps, and they would have a HACCP plan to cover their steps. You will definitely want to audit your manufacturing partner to ensure they're following the correct procedures, and have a Quality Agreement in place outlining who is responsible for which steps... At my company, we use contract manufacturers, and we have a Quality Agreement that is essentially a checklist covering all the GMP/HACCP requirements and assigning a responsible party for each item. For example - Raw materials: are you sourcing materials from your own suppliers, or is your manufacturing partner sourcing them for you, or a combination of both? Who provides the raw material specifications? Who is responsible for acceptance/testing of raw materials? Who is responsible for approving the suppliers? Etc. For raw materials you're providing, you would conduct your own hazard analysis, but you would not conduct a process hazard analysis if manufacturing is done by your partner, because that will be specific to their facility/equipment/PRPs. They would need to complete the process hazard analysis, and you review it when you audit them.
Regarding your HACCP team, if you have a VERY small company and no more than 2 people are relevant, it could be acceptable. If you have any manufacturing at your location, you must have enough staff to include more people (a representative each from management, quality, operations, sanitation/maintenance, etc.)
Hope this information is helpful!