There is no synchronization between the course on the computer (on the site) and the application on the phone.
Not in my phone, not on the phone of my friends.
There are several threads about syncing. It’s not exactly clear to me about which situation you are talking.
Situation for example: self-created course, and downloaded to your phone. After you making a change to your course (for example update the spelling of a word) that change is not reflected in the phone.
This was reported this long ago, but it seems that one needs to delete the download and re-download the course again. It’s not clear to me whether this is intended behavior or a bug (with workaround).
Another situation: self-created course, not downloaded to phone. Again a change is made (such as a spelling update). That update is not reflected on the phone.
My experience is that this goes away if you move around a little, such as go to profile and back to the course, or restarting the app (after killing it). I noticed that it will show the screen with the rocket and that the user needs to wait relatively long before the course levels are shown. It seems that it updates the statistics and learning progress at that time (but I don’t know that for a fact, just deduced based on behavior).
Other users reported logging out and logging in again solved the issue too, but I don’t like it that all courses need to be re-downloaded again; I like small steps over drastic ones.
Another situation: any course, have studied on the website, but learning progress is updated on phone, e.g. you needed to review 23 words (the web and phone show the same number), which you did on the web. Then going to the phone app, it still shows the same 23.
Again, moving around a little bit solves it. Or just learning or reviewing one word, then exit the session with the cross top-left, will also update the progress, I noticed.
The issues above would apply mostly to user-created courses, as Memrise courses don’t have so many changes and the user wouldn’t be immediately aware of them.
It seems that when a course a user is studying and it’s progress is (partially) cached, which is only updated when certain events are triggered.
It also seems that a downloaded course is indeed fully downloaded, and is never updated, unless one deletes the download and re-downloads the course. The learning progress is however nicely updated now and then.
All the above is based on my experience, as there is not exact description of how it is supposed to work.
PS: I have moved your question to the ‘how-to’ section (or should it be the Android-bugs section?) as it deal with using the web/app, not specifically American English.