"A few members have reached out to me to say that their account has been blocked. I have gone into their accounts to "enable" their user status and change their password."
Happy to help a fellow Joomlian Civi user!
CiviCrm generates "contacts". To give a contact rights to edit their own personal contact data you have to give them a Joomla "user account".
A user account is always member of at least one Joomla "user group". Users get assigned the rights that belong to the usergroup they are part of.
For instance: the right to see their own personal data and to change these. Members of a Joomla usergroup that functions as a CiviCrm admin group will have (lots of) rights beyond these two rights.
To set all possible CiviCrm rights for all Joomla user groups you have to login to the site management area (website.com/administrator) as a member in the Joomla superuser group.
After you login, you click on System/Global Configuration (left top-menu in admin area).
You will see 5 tabs with settings on top and left of these a column titled Component.
In this list you click on CiviCRM.
You see two tabs: Readme and Permissions.
In the Permissions tab you can set all rights for all user groups to dozens of CiviCRMs view and action rights.
This permits lots of fine detailed settings for lots of groups. The Joomla rights system is very flexible but takes some work to get familiar with.
I suspect that your problems could be the result of errors in these settings.
Hope this helps you along!