The privacy policies for individual apps can be found here:
Neither the app itself nor the server implementation collects any privacy-sensitive data. However, the app uses some 3rd party libraries which may collect some data. The next section describes libraries, services and permissions used by the app which may impact user's privacy:
1. Google Analytics for Firebase: I use this to get some statistical data on how the app is being used. For instance I see how many users used the app every day, what Android versions were used, what countries are the users from etc. These are only aggregate statistical data and I can't track individual users based on this - I only see e.g. the number of users coming from the United States, the percentage of users using Android 4.4 etc. Almost all web pages use Google Analytics so while you are browsing the web, similar data is being collected too.
2. HockeyApp: This is a crash reporting service owned by Microsoft. As users typically don't report crashes (or at least don't report them in a way that is useful for me), I use this service so the app can send a crash report whenever it crashes which helps me locate where the crash happened and fix the issue.
3. Location information: When the location permission is enabled, the app uses it for two purposes - to find the nearest servers to test against when performing the speed test and to plot the device's location in visual traceroute. Both of these features are available in the paid version only. In addition, the app is required by Android 6 SDK to have the location permission when accessing the wifi signal information - the location information isn't used in this case at all. See this article for more information.
4. AdMob: This is a Google-owned ad-serving service used by the free version of the app only.
I understand that because the app shows quite a lot of low-level information about devices and wifi networks together with their MAC addresses and asks for the location permission, it might be scary what happens with all the data. I could be an evil guy who sends the data his server for his evil purposes. However, when you think about it, any app can do the same (even access wifi networks without the location permission). So if I were an evil guy, I'd create a nice game with cute kittens children and parents would love, run all the scans in the background and send the results to my servers - while users are filled with joy how cute the kittens are. Actually I'm wondering how many apps like that are on the Play Store...
Finally, notice that there's no place in the app where you would enter any personal data such as name, email, address, phone number etc. and the app doesn't have permissions to access sensitive information such as contacts, photos, camera, storage, etc. So even if you consider the worst-case scenario that I am lying in the above and store all the information somewhere, there's no way for me to relate the information to you as a person. All I would see is that e.g. someone at the given location used the LAN scanner and know the IP and MAC addresses of the discovered devices. But as I wouldn't know who the person who did the scan was, I would be very limited in how I could misuse the information.