The architecture of the project must be well organized according to principle "low coupling and high cohesion"
The system layers should be well seperated. The basic architecture have 3 layers.
- Presentation Tier: JSF managed beans are in this layer. The task of this layer is to correspond user actions, hold user data and show the data to the user. Basically, its responsibility is bounded by MVC pattern regardless of which implementation you use (JSF, Spring MVC or Struts)
- Bussimess Logic Tier: The data and actions which was collected from page was sent to this layer to operate on data. You should not manipulte the data and prepare the response inside Presentation Tier. It is the responsibility of Bussiness Logic Tier.(Spring, CDI)
- Integration Tier: Database access should be done in this layer regardless of the
library technology used (Hibernate, JPA, JDBC).

As you can see in the pic. your JSF managed beans should only be capable of requests from the pages(controller) and page data (model). You can have a look to this answer to understand JSF MVC. Therefore, It is better to not to connect DB inside JSF managed beans. It is not a good practice.
The second part of your question. All technologies, Hibernate, JPA, Spring JDBC, uses JDBC driver of the related DB. JDBC Drivers is the just thing which connects to the DB. However, it is beter to select a method how to connect to DB, Hibernate, JPA or Spring JDBC.
You can download my example application which implements this architecture basically.