What happens if two I(1) series are combined that DO NOT share a common trend?

If two I(1) series do not share a common trend, then combining them will likely result in an I(2) series. An I(2) series is a series within which the second diffrence is stationary. This means that any two values within the series will be statistically independent of each other, meaning that the influence of the first series will not be visible within the combined series.