There are no separate access permissions to activities in Bitrix24. You need to configure access permissions to a CRM element which activity is related to.
For example, if an employee has access to deals, he/she has access to activities related to deals as well.
How it works
Click CRM > Settings > Permissions > Access permissions.
Let's configure permissions for Manager role.
For example, let's configure access permissions to deals. Access permissions to companies, contacts and leads are configured the same way.
Give access to Read action and select All.
Now, users having the Manager role can view activities related to any deal.
He/she won't be able to edit or add new activities and will get an error message instead:
Let's give Manager role an access to Update action.
Users having the Manager role will be able to edit or add new activities related to any deal:
Access to Delete action works the same way.