Before assign a task to a person, you need to add them to the project and a board (if it's private to a project). As there can be different departments in your company and each of them has its own working project and boards. Let’s say there’s no need for an HR-specialist to be on Web Development board.
So once you have invited all members to the company next step is to add them to the project:
Open project’s settings
2. Open Members tab > Add Members
3. Open a board, go to its Menu
4. Add members
5. After that you can add a person to the task