I understand that when I empty the recycle bin, the files are still within the computer, and would be accessible by someone with special knowledge. My question is this: the next time that the recycle bin is emptied, would the original deleted files be completely erased?
It will depend on the type of disk you have. On a HDD, the files will be marked as deleted in the Master File Table (MFT) but the actual data is still there. If you do not write anything on the disk, programs like Recuva are able to recover the data.
If the disk is a SSD, the mechanism is different. The minute you delete the files from the recycle bin, Windows (7 and up) will signal to the SSD controller that those files are deleted (Trim command). That will erase any trace of the data on the SSD. The MFT will be handled like in the case of the HDD.