However, we don't know what the effect actually looks like over a smattering of seasons. How much better than average for the win-streak teams? How much worse than average for the losing-streak teams?

But, wait, that's not right...we have to subtract out the ten games and go from a .500 value that's ten fewer games than the full season. 162-10 = 152, so: 76-76. Add ten to the win column for the win streak: 86-76. Add ten to the loss column for the losing streak: 76-86.
Don't like the .500 assumption? You're probably quite right. Losing that many games in a row is not something that a .500 team is very likely to do. Our chart of the past 13 seasons over at the left (which you've probably peeked at already...) shows that of the forty teams who've had a 10-game winning streak, only five wound up with a losing record during the year in question.
And of the forty-six teams that have had a ten-game losing streak (there are three teams who've managed to have two ten-game losing streaks in the same season: all three of these teams, as you'd expect, lost 100 or more games during the year in question) only four of them had winning records.
So the first approach--subtracting from 81--actually gets closer to the real-life totals. The aggregate won-loss record for teams who have a ten-game winning streak is .553, or about 89.5 wins per 162 games. The aggregate won-loss record for team who have a ten-game losing streak (to be accurate, that's ten or more consecutive wins or losses) is .422, or about 68.5 wins per 162 games. (This would go up to close to 71 wins if we removed the double-dippers.)
And that gives you some more specifics about something that you were all experts about, generally speaking, before we went off and crunched the numbers just to hear them squeal in terror and delight, as good numbers should always do when they are given a good pinch.