There is a sociological term called "collegiality". Strictly, it means the relationship between colleagues. The idea is that a college is a group of people united for one purpose. This usage of "college" can be seen in "Electoral College" or "College of Cardinals".
The idea behind collegiality is that while there is a separation of individuals, all individuals present come together in one mind and for one purpose.
This is similar to the idea expressed here in John 17:11 from the question.
What Jesus is praying for is that we will share his thoughts and concerns and that he will share ours. He's praying for collegiality: that we come together with one mind and for one purpose.
Placing this in relation to John 10:30 becomes tricky since some denominations see this as collegiality while most see this as scriptural support for the trinity. A survey of that is a bit beyond the scope of this site (being doctrine).
However, there are other references to collegiality in the New Testament including this one:
Phillipians 2:2 (NASB)Emphasis added
2 make my joy complete by being of the same mind, maintaining the same love, united in spirit, intent on one purpose.
The oft-cited verse 3 goes on to a bit more practical application of collegiality:
Phillipians 2:3 (NASB)
3 Do nothing from selfishness or empty conceit, but with humility of mind regard one another as more important than yourselves;
When Jesus is praying that we "may be one", he is praying for our collegiality--he's praying that we will be of one mind and united in spirit for one purpose.