How to set up a multi-store with different products in Magento?

Question

Magento

Is it possible to set up 2 stores on different domains, both with individual products and general ones?
The basket should be common for all stores, search too, when you go from search results in Store#1 to a product from Store#2, the URL corresponding to the second store should change.
Tried several configurations.
1. One site, 2 stores with their own "shop views" on different domains. In fact, we got the main pages on different domains, and the products are all together.
2. Different sites - everything is separate, nothing in common.

Answer 1

Find a programmer. Let it do it for you.
This is a non-standard task, and Magenta is intended for typical ones.

Answer 2

For each store, you need to configure your root "Category"
Done here Stores > All Stores > Store: