Have you checked the footage interpretation and enabled aspect ratio correction in the comp/ source/ layer viewer in AE? Generally AE should correctly handle the clips just like Premiere since it uses the same routines, but PAR correction is off by default in AE, so it may simply look like 4:3 in teh UI...
Yes, this is what I tried before writing this post because it seems to be the solution (according to some forums), but I couldn't manage to do it. It may have happened because I'm doing something wrong, so could you please give me more details about what I should do to check that? I mean which panel, which option, and so on. I hope it won't be hard or something. Thank you so much!
At the bottom of the Comp Window you'll find the pixel aspect ratio compensation toggle switch. Toggled on, everything's in the proper screen aspect ratio, but the edges look stair-steppy, expecially text & graphics. Toggled off, everything is smooth but not in the proper screen aspect ratio.
When rendered, everything will be in the same screen and pixel aspect ratios as your comp, AND it will be nice & smooth.