Ability to setup email and custom notifications is available version 4.129 onwards
Twilio for Salesforce allows you to send email and custom notifications for incoming messages without the need to setup any process builders or flows in Salesforce
Notifications can be setup under Notifications section on Twilio Configuration Page in Salesforce
For messages received on a Personal number, you can choose to notify the owner of the personal number or the owner of the record(lead or a contact or custom object) to which the message belongs
For messages received on the numbers in the Default Messaging Service, you can choose to notify the record owner
Allows you to push chatter notifications to the personal number owner
Setup the default sender and content for chatter messages
Please note, the Notification Templates, and which Agent to Notify settings are not applicable for chatter notifications
We encourage the use of custom or email notifications as opposed to chatter notifications because of better customizations.