The answer is both yes and no, with the majority of the time being "no".
First, if you build a desktop website most web developers use "Responsive" themes. These are themes that automatically produce a web AND desktop version from a single source (without needing to produce additional code). This is the most used way to conquer needing to develop for mobile. These themes can be obtained for free if you use wordpress.org, or purchased for a nominal fee (usually 35-75$) through theme websites.
Also, certain startups - like those of Evernote - revolve around Apps found in phone app stores. Evernote receives MOST of it's mobile traffic through the app store. Likewise, the gross majority of those who visit the website do so by desktop. This means the mobile (app) version and desktop versions are how they are because they are built for their primary incoming traffic.
The only time you will see a need to develop for mobile is if your desktop site is specifically designed for a large resolution monitor and you cannot/will not change it.
Hope this clears things up.