It may be worth spending a little time setting up the ticket system before you use it. There are several aspects to setting up the ticket system.

Department Ticket Configuration

Firstly, double check the department configuration. You can find this in Configuration / Departments (log in as admin first to be able to access this). 

Those involved in a ticket are called "ticket participants". There are three ways that users get involved in tickets:

1. The user who created the ticket will always be added as a participant.
2. Each department you add will have an area underneath it for Default Ticket Participants. All the users listed there will be added as a participant under a ticket.
3. When you edit the department record (or add a new one), you'll see an option for "Default Ticket Assignee". This is optional, but if you set it then the user you select will always be assigned the ticket. This is a "strong participation": not only will this user be added as a participant, but it will be assigned to them through the workflow engine. This means it will be highlighted in red and will remain assigned until the ticket is closed or reassigned to a different user.

Ticket Access

Ticket access is based on a couple of configuration options:

• Whether the user has the "Only Sees Their Tickets" option set in their user profile, in which case they can only see tickets that either they created themselves, or that were assigned to them in some way. This is the default option. 
• The department they are in. If the "Only Sees Their Tickets" is not set, the user will see all tickets for their department (ie. the department set on their user profile). This means all tickets that were set to created with that specific department set. Otherwise the user will be able to see all tickets in the system. Note: The user must also have their Primary Role set to "Help Desk", or they will still be limited to see only their own tickets.

You can also override the departments that a user sees if they don't have "Only Sees Their Tickets" set. You can use the "Set Department Filter" button on the user profile record. This action is only available to administrator however. This lets you override the departments that they see, and also lets you specify more than one department. Then the user will see all the tickets for all the selected departments.

Note: You may have to log out and log back in again to see these changes, as certain permissions like this will be cached for performance.

In addition, if a user is set as a Default Participant for a department, or they are set as the Ticket Default Assignee, then they will always be able to see new tickets for that department.

In general, you should add the users who will be fielding new tickets as default participants. That way they will always receive new tickets and can then decide who else to pull into the ticket. Read more about this in Receiving a Ticket.