You need to assign users to group then allow that group editing permissions in K2 categories.
1. Check K2 Front End Editing is enabled.
Go to joomla admin > component > Select K2
Look on your right and click on "Parameters"
Click on Front Editing Tab
Enable Front Editing. This is a global setting.
2. Add Users to K2 Groups.
Go to K2 > Users and make sure they are assigned to your required K2 Group. In this example I will use " Reporters Group".
So I click on user "Reporter1" and assign to "Reporters" group. If you haven't got your groups set go to step 3 then comeback to this step 2 again to assign your users to groups.
3. K2 Groups.
By default K2 has registered and Site owner groups. Depending on your site you might want to create more groups.
So simply create your groups and check settings. On top you have permission settings , below that you have K2 categories where this group will have permissions.
So i create "Reporters" group and allow permissions to K2 News category.
Now when a user logs in they will be able to edit K2 articles for their assigned categories.
You might need to create and use K2 Login module which has extra links to create or add new articles.
See you around...