Goldman joined the company as vice-president of product in February 2007. Before this, he was product manager at Google for more than three years.
His departure follows that of chief executive Evan Williams, who stepped down to focus on product strategy at Twitter.
Williams, who co-founded the site with Biz Stone in 2006, has been replaced by its chief operating officer Dick Costolo.
Earlier this week, Twitter announced that it was adding more multimedia content to the site and has partnered with five new media companies, including web TV company blip.tv and presentation-sharing site SladeShare.
The company said it will be announcing more content partnerships in the coming months.