When I run a container as a normal user I can map and modify directories owned by root on my host filesystem. This seems to be a big security hole. For example
Access to run docker commands on a host is access to root on that host. This is the design of the tool since the functionality to mount filesystems and isolate an application requires root capabilities on linux. The security vulnerability here is any sysadmin that grants access to users to run docker commands that they wouldn't otherwise trust with root access on that host. Adding users to the docker group should therefore be done with care.
I still see Docker as a security improvement when used correctly, since applications run inside a container are restricted from what they can do to the host. The ability to cause damage is given with explicit options to running the container, like mounting the root filesystem as a rw volume, direct access to devices, or adding capabilities to root that permit escaping the namespace. Barring the explicit creation of those security holes, an application run inside a container has much less access than it would if it was run outside of the container.
If you still want to try locking down users with access to docker, there are some additional security features. User namespacing is one of those which prevents root inside of the container from having root access on the host. There's also interlock which allows you to limit the commands available per user.