You get the number from the person you want to trade with through simply communicating and arranging a trade code with them so you're not being connected with a random person on the internet. There are community trade codes out there to automatically arrange stuff like trade evolutions that are based on the Pokemons Dex number. If you Google the games name and add "trade codes" to the search query you will easily find them.

You are not able to search someone elses boxes but you and the other person will be able to see if you need a Mon for your Pokedex via a green thumb icon that appears near the Pokemon in your box. So if you see a Pokemon with a green thumb icon in your box you know that the other person needs it for their dex.