By learning web designing on one`s own, how long till we can build our own website?

First to answer to answer your question: I don’t know how technical you are, but if you study diligently, you could probably learn how to build the layout in a few months.

But then there is the other side of learning how to build the databases etc to contain the information you need.

Second: Why don’t you just use a service like Ning which are already built so that people can build communities.

If your goal is getting the site running (rather than learning how to build it), I would recommend becoming expert at a Content Management System (CMS) instead.

Security, profiles, forums – all that has been done before.

Some popular CMSs:
Drupal
Joomla
DotNetNuke

I’m with @jaytkay – Don’t bother coding it yourself. Why reinvent the wheel?
IMO Drupal is the best CMS / framework to start with as a beginner. It offers a good mix of ease of use and flexibility. Also it is well documented.

Here’s a more direct answer.
If you insist on coding it yourself, it will take a long time. How long depends on your talent for it and how dedicated you are.
Learn HTML first. Learn CSS next. Learn Javascript after that. Get to know these two languages reasonably well before proceeding. You probably won’t do much hand-coding of HTML or CSS once you get rolling, but you need to know them. HTML and CSS are the languages that are responsible for things displayed in a Web browser, so when you server-side scripting later you will be concerned with things that output HTML/CSS and Javascript.
HTML encapsulates the actual content of a given page.
CSS determines how the content will be formatted and displayed (style sheets).
Javascript takes care of “local actions” on the page (generating new local content without calling back to the server).
After you have a handle on the client side of things, it’s time to learn a database. SQL is SQL pretty much everywhere you go; there are minor differences but the query language is pretty static AFAIK. Most websites use MySQL, Postgres, or MongoDB as a database backend. MSSQL and Oracle are up there too (in fact they might be more popular than the other three, but I don’t hear about them in my “neck of the woods” – FOSS programming).
The database is where all the data for generating your site will be stored. It is much more important to have a good understanding of how the database works than to worry about the peculiarities of a server-side scripting language, and that’s why I recommend tackling it first.
Unfortunately, there’s relatively little you can do with your database without said scripting language (except that Javascript can access certain types of properly-configured databases) so it is tempting to try to learn both of once. I made that mistake myself, and I warn others against it.
Finally, you will need to learn a server-side scripting language. PHP is far and away the most popular of these, but far from the best IMO. Ruby and Perl are also popular. Python was once in wide use in this way but from what I understand it’s no longer usable without the Django framework (i.e. mod-python for Apache may be deprecated.)
So, it’s a lot to learn. These are the minimum things you will need to learn to code a site like that yourself. You would probably also need to learn to install, configure and administer your own web server and database engine unless you have a competent sysadmin available.

if you need some help…. there is always
http://www.kickstarter.com/