What does a Sunni want from a Shi'a?

Answer 1

They both want from each other to understand that they are both Muslims who are:

• believing in one and same God
• believing and following same prophet, Muhammad (PBUH)
• reciting and memorizing the same holy book, Quran
• praying facing same direction (Kibla), Kaba
• Performing pilgrimage (Hajj) to same place and same ritual worship
• fasting same month, Ramadan
• believing in the Day of Judgment
• believing that they differ only on minor Islam issues but never on basic Islam principles and teachings.
• believing that they should cooperate for the benefit of their families, countries, religion, and humankind.

Refer to related question below for more information.

Answer 2

This depends from the Sunni in question. You have moderate Sunni Muslims, like fellow Expert Ibrahim El-Osery (whose view is also hre), who want a cordial friendship and unity in Islam between Sunnis and Shiites, and you have more fundamentalist or Jihadist Sunnis who want Shiites to either "convert" to Sunni Islam or die. In between, you have a large number of Sunni Muslims who live uneasily with Shiites and would like to trust their Shiite neighbors not to hurt them a little more.