I would suggest moving products to a separate collection.
MongoDB data schemas are best when based on the usage requirements. In this case, a user browsing products is going to very rarely be interested in all the products that a different user is selling. Usually, they're going to be interested in products that match search criteria about price and item description or item title.
So imagine a search for the most recently posted items given your proposed data schema. Your query will have to search through every user's array of product subdocuments and check the timePosted. Whereas a product collection could easily have an index on timePosted and not require touching unnecessary (user and each user's other products) documents.
The use case for seeing a single product page is also important to consider and also indicates using a separate product collection.
Even when you do need to show products that a particular user posted, it will not be difficult in a product collection to search by user_id.