Publish will make a folder (or file) accessible to anybody. No password or logon is required. This is used to create static links to your files. You can use the static links on any webpages or any websites, e.g. eBay, Facebook, etc.
Published folders / files are always read only. This is because any users can access published files / folders. However, to upload / delete / modify files on DriveHQ.com, DriveHQ membership is required.
To share a folder to friends with read-write permission, please use the Share feature, not the publish feature.
You can share different folders to different users with different levels of access rights. The other user also needs to have an account on DriveHQ to upload files (to a folder shared with full-access or add-upload right). If the other user doesn't have an account on DriveHQ, he/she can sign up a new account and then gain the right access level to the shared folder.
A group account can make the process much easier. The group account service is free and offers one free sub-user license. To become a group account, just logon www.drivehq.com, go to My Account --> Group Account.
Once you have a group account, you can easily create a sub-user account for your friend(s)/customer(s). Your friend(s) can logon as the sub-user account and access the folder(s) you shared to the sub-user.
For new users, the easiest way to share files is always to give your own username / password to the other user(s) so that you can logon as the same DriveHQ user.