Well...I wouldn't necessarily go by just shine/no shine...though yes, generally the dull side would indicate some kind of tape backing.
Of course, not all tape has that extra backing layer, so it might appear shiny on both side.
If this is a new/unused reel of tape...with the top/front/label/logo side of the reel facing you, new reels are always factory wound to be placed on the left/supply side of a tape deck (at least I've never seen a new one wound for use on the take up side). So if you put it on the supply side, that should be the way it goes, regardless of shine/no shine.
Of course...if someone has used it or spun it up on, the tape direction might have changed. IOW...many people (I am one) will always take a new reel, flip the whole thing upside down, so the top/front/label/logo side is now facing away from you, and then put that reel on the right/take up side...then RW to an empty reel on the supply side....and then flip the empty on the right/take up, over, so the top/front/label/logo side is again facing you.
Why...?...(you may already know)...so that when done tracking, you always wind it back to the reel it came on, which is now on the right/take up side...so that it gets stored "tails out" to minimize any effects from print-through.
Maybe that was too much information for you...
...so the simple thing is to just record the same thing on each side of the tape...shiny and non-shiny...you will be able to tell by the sound quality on playback which is the correct side to record on. The wrong side will sound kinda dull and maybe lower in level than what you put in.