You want to create a sharing set. (taken you're using customer community)
In theory, you could provide Public access for either Cases or Contacts. However, doing that would almost never make sense for an organization. Think about it — you would be providing each of your customers visibility into all Contact and Case records for all of your other customers. Assume your Organization-Wide Default for both objects is Private.
To mitigate the deliberate, but potentially debilitating, gap in the sharing functionality between Customer and Partner Communities, you can create one or more Sharing Sets here: Build > Customize > Communities > Settings > Sharing Sets. Scroll down to the “Sharing Sets” section and click “New”:
The first key piece to understand about how Sharing Sets work is that no more than one Sharing Set can be created for each available profile. You cannot create multiple Sharing Sets that provide different access and apply both to the same profile. Go ahead and set your label and description for your Sharing Set. Next, select one or more profiles for this Sharing Set to apply to.
On to the fun part. Here, you’ll need to select the applicable object. In this case, you’ll want to select Case and move it to “Selected Objects”. Go ahead and click on “Set Up” next to Case and the Access Mapping for that object will appear:
As I stated before, the going assumption is that you want to provide Read/Write access to all of the submitting Contact’s colleagues. To do this, you will want to share the Cases to which all Contacts on the submitting User’s Account has access. In other words, provide access to all Contacts where his/her Account = Case.Account. Here is the configuration:
Say for example you wouldn't want to share all cases logged on the account of the user which is currently logged in in the community, but just the cases who have the contact lookup filled in, and the chosen contact has the same account of the logged in user, this would be your configuration.
Credit to Phil Weinmeister of EDL Consulting for the nice blogpost