Could anybody tell me what is the name of the place or how can I find the additional info?The name is Osawa Onsen in the Hanamaki area of Iwate Prefecture. More information can be found on the following page:
http://www.japanican.com/hotels/shisetsudetail.aspx?st=2335001
Or are there the similar hot springs like this? I am looking for a private ryokan for two people.
Yes, but not many of them are private. Among the best (non-private) riverside baths I peersonally have been are the above mentioned Takaragawa Onsen in Minakami, Gunma Prefecture and the baths of the Yamamizuki Ryokan in Kurokawa Onsen, Kumamoto Prefecture. But there are many more.