Who is your host and what is your website about?
- you process payments
- deal with sensitive political, social, or religious topics
- host what might be called "adult" material in some jurisdiction somewhere
- do anything involving minors (games, clubs, etc.)
- allow users to post files or photos
- keep any personal information or records in a private directory on your host's servers
...then only deal with a well-known reputable host provider. They have the infrastructure and security assets to help keep you out of preventable trouble.
If it's just a personal website or blog that doesn't have heavy traffic, or get spiked on a regular basis, it shouldn't matter. Take a look at your traffic logs. That should tell you if you need to move up to dedicated or virtual hosting from shared.
As Shades pointed out, the decision to get a dedicated server is more of a traffic and uptime issue than a security one.
The ultimate quality of your site's security is determined by your host. You can be just as vulnerable
with a dedicated server as you can with a shared if your host isn't overly concerned about security. The other part of the formula is YOU. If you're sloppy and careless with your passwords (or configuration settings on a dedicated server), the most secure host service in the world won't be able to protect you.
To recap: if you're concerned about security - get a different host. If you're concerned about traffic, uptime, or quality of web experience - consider a dedicated server.
P.S. Welcome to DonationCoder!