I do a bit of both. It's important to know how to manage my own webserver, so I do that for certain projects. Most of the time I let someone else handle it so I don't have to bother with updates, hardware, security, uptime, logging, monitoring, etc. Running your own webserver is a great experience but it is definitely time consuming.
In an article I wrote about running a webserver , I pointed out that most people don't need to run their own.
Most people want to upload a website once and forget about it for a few years. If you're running your own webserver then you can't really do that.