If I go private label, do I actually own my formulation, or does the factory?
On pure private label, no, you're customising within the factory's existing base, and they usually retain rights to that base formula and can sell variants of it to other brands. If IP ownership matters to your brand story, you need a contract-manufacturing agreement with an explicit IP clause, in writing, before you share anything; a verbal 'this is exclusive to you' means nothing in a dispute.
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Focused specifically on formula ownership and control, the exact question founders ask once they realise 'exclusive' doesn't always mean 'owned'.