Right now everybody have access to the info of all the contacts. And we need that only specific people of our team get access to the info of a specific group of people.
How do we do that?
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ACL or Access Controls should be able to do that. https://docs.civicrm.org/user/en/latest/initial-set-up/permissions-and-access-control/
BUT... I've never worked with Joomla so can't offer any further advice
You can create ACL through UI if you have very limited group if you have 50+ groups and will increase going forward than would recommend to write a small extension that implements ACL civi hook to control the action on contact.
Some useful docs to configure ACL via UI:
For some use case you may like to use below extension