Hello, Peter. Yes, you will need to do some extra work to make sure that your two street datasets actually connect properly to one another. In order for streets to actually be connected in a network dataset (ie, it's possible to drive from one to another), they must have either endpoints or vertices at the locations where they touch, depending on your network's connectivity policy. If they simply cross over one another without vertices or endpoints, they will not be considered connected in a network. It will be treated more like an overpass, where no connection is possible. You can learn more about network dataset connectivity here: Understanding connectivity—Help | ArcGIS for Desktop
One of the easiest ways to ensure connectivity is to set your connectivity policy to Any Vertex and to use the Integrate tool with an XY tolerance of 0. Integrate is sort of a scary tool that edits your input features in unpredictable ways, but the 0 XY tolerance ensures that nothing will actually change shapes. It will just add vertices at the points of intersection. It also honors the selection set, so you could first select only the small number of features that you're trying to connect before running the tool.
