Sounds like some kind of spy/infiltration related slang. But friswing suggests that's not the case in Swedish.
I've nicked this explanation from someone called Brian Hitchcock on Stackexchange:
Some things have an "inner" suface—a window pane, for example, has a surface that faces outward, and a surface that faces inward. They have an inside (ie, the "in" side) and an outside (ie, the "out" side). So it makes sense to say the inside of a window. Other things (e.g. an apple) have an "inside" that is entirely contained in the thing. The inside of an apple is everything but the skin.
I.e. condensation on the inside of the window is on the inner window pane, but condensation inside the window can be between two panes.
I think in daily life people seldom say "on the inside". I can only find one explanation for the phrase "on the inside" which is "If someone is on the inside, they have a job or position in which they have special or secret information". Betyder "på insidan" verkligen det?
No, it does not. If that is the case we can use the English 'insider'. 'På insidan' can be in a house, in a box, a page in a book or newspaper ... "inuti" is the best choice when talking of what is "in a box". "På insidan" feels more like in a book, newspaper, or clothes of somekind (the side touching your body)
"In the inside" absolutely unacceptable? Is this an English problem or a Swedish one?
In English I have only heard 'On the inside...', which corresponds to the Swedish "På" in this context.