When I have implemented CiviCRM for various clients I have always set it up on its own server. One of my clients saw some benefits in having it run on their website server - partly economics - and so it was moved. And then the problems started...one problem was conflicts with the many Drupal modules on the website which broke various bits of Civi. So my conclusion is that it's better to run Civi on a dedicate server where the environment can be controlled & suited to Civi. What do others think? What are the pros and cons?
I have worked with many clients that maintain their website (usually Drupal, but WordPress and Joomla as well) and CiviCRM running on the same server, usually in the same domain instance. I seldom have seen issues.
Now, of course, with a mis-sized or misconfigured server, you will have problems. And if the website gets a huge amount of traffic, you will have problems. And if your CiviCRM database has many hundreds of thousands of contacts, you will have problems. But these type of installs are few and far between.
There are definite benefits running a Drupal website and CiviCRM together. You are easily able to embed contribution pages and event registration pages into a Drupal site. With Views Integration to CiviCRM you can pull information out of CiviCRM for display on the Drupal site. With Webform Integration with CiviCRM you can build robust webforms in Drupal updating CiviCRM.
Ultimately, the key is properly sizing and configuring the LAMP stack, Drupal, and CiviCRM. (Additionally, try to install on a VM rather than a VPS. You want full access (ala root/ssh) to tune things up.)
It depends what you mean by "server" and "website". A single server can have multiple websites, with (mostly) their own environments. It sounds like it was not only put on the same server but installed as a module into the same website, i.e. into the same set of folders as their "main" website.
If the hosting plan allows for subdomains, you could install the civi "website" into a subdomain on the same "server", and then they would be separate. You will likely have to share the RAM, php version, mysql version, and apache (or nginx or whatever) version, but otherwise you can then set up the environment as you like.
So the main factor is whether php/mysql/RAM on the desired server meet the minimums required for Civi.
I agree it depends, How active is the website and how active is Civicrm. I am running fine on a Cpanel with 2 additional test systems and a few other accounts. We are using Drupal and everything plays nice.
Civicrm is very database usage heavy. Depending on your server 1 transaction a second may be a light load.
Do a little math to understand your load and ask the question, somebody should have some benchmarks which can give more meaningful answers.