There are some good answers here which cover a lot of good points comparing the two. From my personal experience, I would strongly recommend an online editor like WriteLaTeX or ShareLaTeX for beginners to learn with. LaTeX has quite a steep learning curve at the beginning, and compilation is not always as simple as you might expect (it might be necessary to run LaTeX twice to compile a document, or run LaTeX, then bibtex, then LaTeX again....). The online compilers handle all that for you, so you can focus on learning LaTeX itself. Things like live previews showing when you've made errors are also really useful when you're starting out.
A recommended reason for using a local install is that it gives you more control and customisability. This is something that's likely to come later — not when you're just starting out. (And that said, the online editors tend to have most of the packages you are likely to use installed anyway).
When you are past that beginner's hump and starting to itch for more customisability, it's useful and important to learn how to set it up on your own machine and get everything working that way, too.
In the long run, unless there is a reason you can't use the online editors (maybe you need some package they don't support), I think it comes down to your individual circumstances:
Do you need to be able to access and edit your documents from different computers, including maybe a public library computer? Then use the online editor.
Do you always have your laptop with you, but not always have internet access? Better to install your own copy.
Are your documents extremely high security? Best not to put them on the online editor's servers.
Are the online editors sufficient for your needs, and are you collaborating with others? The online editors have great capabilities for google docs-style real-time collaboration.
At the end of the day, it comes down to what you find suits your needs best — if you're not in a huge rush, why not try an online editor for a couple of documents, then try a local install for a couple of documents, and see how you find them?