You need to write a sharing rule to open up the access rights. It can be criteria or ownership based as per your use case.

Provided that, the user's profile/permission set must have Edit access to Contact object.

Refer [Define Sharing Rules][1]

Here is a way to open up visibility & access rights

[![Sharing architecture][2]][2]


  [1]: https://trailhead.salesforce.com/en/content/learn/modules/data_security/data_security_sharing_rules
  [2]: https://i.sstatic.net/6dQns.jpg