Question: Should I use a dedicated server or go with a managed Drupal web host?
Answer: If you have experience in managing a server then you should use a dedicated server. If you do not have experience in managing a dedicated server fine tuned for Drupal then you should look into getting a managed Drupal web host.
Drupal managed web hosting will require the web host to fine tune your server and provide additional add on services. If you go with Drupal managed web hosting you do not have to worry about server issues while you can focus on managing your Drupal sites.
If you decide to run your own server. You will need to dedicate time to managing the server and troubleshooting any issues that may arise.