It is not unexpected to get different results with ArcGIS Online and your own network. The network dataset used by ArcGIS Online is a comprehensive dataset using the highest-quality data available from our data vendors. It includes detailed street and path geometry as well as an extensive set of attributes defining the characteristics of the streets, which are used in calculating impedances and restrictions.
Your own data may have some differences in geometry - roads in slightly different places, certain roads missing, etc. Thus, the optimal paths between places on the network may differ slightly. You might also have some connectivity errors - places where the streets seem topologically connect but aren't actually logically connected in the network.
Even if the geometry of the streets is identical, there is probably a difference in attributes between your data and the ArcGIS Online data. The impedance calculations may be different, and ArcGIS Online might be using some restrictions that you are not.
Finally, it should be noted that the ArcGIS Online data contains information about turn restrictions - specific turns that are not allowed or which incur a delay. If you use the "Driving Distance" travel mode, these turn restrictions will be used, even though, for your use case, this probably doesn't make sense. If your own network doesn't have any turn restrictions, it may allow you to build your pipes following the streets around a corner where vehicles are not allowed to turn, like an intersection where you cannot turn left. But for pipes, I presume this doesn't matter. Thus, your data may actually be more appropriate for this situation. Note that there are ways you can adjust the travel modes even when using ArcGIS Online to turn off this restriction, so the issue is not insurmountable.
As far as testing your network, it's pretty hard to interpret the results of a Location-Allocation problem. It might be better to do some Service Areas where you turn on the Lines output. This will help you look for holes in your network where things might be cut off. You can also use Route to test some specific point-to-point routes to see if the results make sense.
If you are concerned about topological connectivity in your network, you can try using this downloadable sample tool to identify topologically disconnected areas: https://www.arcgis.com/home/item.html?id=b2227f745a6c4e1c94dd57810729d2a9