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This sounds like default Magento functionality to be fair. You can have several stores on one Magento installation, and assign an admin to each store so he will be able to make all necessary choices when adding new products.
If you need to specify rights for each admin, this is not default Magento anymore. You will need an extension that allows you to set different permissions to each admin. Take a look at this one, this is a possible solution.
Aitoc is a group of distinguished web developers that have been building for Magento since 2009.
See our extensions for Magento 2.