you could set up an issue security scheme (https://confluence.atlassian.com/display/JIRA/Configuring+Issue-level+Security).
Use a default security level with group A( the group to see all issues) and assignee.
Thank you Udo Brand for your answer. Can you please explain little more?
For example, what security level should be added for group A (group to see all issue)?
What security level should be assigned for group B ( to see assigned issues only) ?
You add a security level to issues not to groups. Within a security level is defined who can see an issue, so in your case you would add to this security level the Group A and the issue role assignee. In order to see an issue with this security level you must be in Group A or you must be the assignee. Please check the documentation provided.
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