Addendum (0:00am, June 27th, JST): After reading Draconis' answer, I did a little more research and added my findings below the horizontal line.
Can medial /t/ and /d/ before syllabic /n/ be easily distinguished? Here is an excerpt from p. 84 from Marianne Celce-Murcia, et. al., "Teaching Pronunciation," 2nd ed., Cambridge University Press, 2010.
... a consonant sound often anticipates some following sound. Both /t/ (in words like but͟t͟on and kit͟t͟en) and /d/ (in words like sud͟d͟en and had͟n't) exhibit unusual medial variants before syllabic [n̩]. The /t/ is either glottalized (articulated with a momentary blockage of the airstream in the vocal cords) or it is replaced by a glottal stop [ʔ], which is a sound formed when the vocal cords close tightly so that air cannot pass between them. When the consonant /t/ and /d/ is produced before syllabic [n̩], the air used to produce the stop is released through the nose rather than the mouth. Because of these articulatory differences, words with medial /t/ and /d/ before syllabic [n̩] are clearly distinguishable, for example, sud͟d͟en versus Sut͟t͟on, bid͟d͟en versus bit͟t͟en.
I am not clear on "these articulatory differences." I guess one interpretation would be that while the medial /t/ before [n̩] has either glottal reinforcement (pre-glottalization) or glottal replacement (borrowing terms from Wikipedia's entry on "T-Glottalization"), the /d/ counterpart does not, and that is the difference. The part about the /d/ is only implied, not explicitly stated.
Is this correct, though? No glottalization occurs to the medial /d/ before [n̩]? When I look up the pronunciation of bitten in American English on Forvo and compare that with that of bidden there, they sound quite similar and the only noticeable difference, at least to me, is the length of the preceding vowel. And even that may be a meaningless observation because these two were clearly spoken by different people. (I know Forvo is not the most reliable resource, but it is the only readily available one for me at this time.)
They also speak of post-nasalization (Wikipedia's entry on "Stop Consonant"), but that is supposed to be a common phenomenon to both /t/ and /d/, so it cannot be a differentiating feature.
I would appreciate it if you could shed some light on this matter. I am not a linguist, still less a phonologist or a phonetician. I teach English (American English, to be specific) and I am trying to educate myself with English phonology/phonetics just so I can be certain of what I teach to my students. One big problem for me is that I do not have innate knowledge of correct pronunciation because I am not a native speaker. Your help would be much appreciated because I honestly do not know whom else to turn to.
Even after reading Draconis' answer, I was not convinced that bidden is read [bɪɾn̩] --- I was fairly sure at least it is not always. Pronuncian.com's podcast episode "66: Syllabic n's and nasal plosions (as in the words 'sudden' and 'couldn't')" deals with this issue head on, and the author says bidden many times in this episode, and she realizes the /d/ as [ɾ] only once, at about 2'30" mark.
The rest of the time, she realizes it as [d̚] (no audible release). You could say it is actually [dⁿ] (nasal plosion/release), but I personally do not think adding yet another allophone is helpful, as far as American English is concerned. Wikipedia states to the same effect.
So in General American, bitten is read either [bɪʔn̩] or [bɪʔt̚n̩] while bidden, [bɪd̚n̩]. Since [t̚] = [d̚], I would say they sound quite similar; to my ears, they definitely do. Wouldn't you agree? I guess whether you can differentiate them or not depends on if you can tell [ʔ] from [d̚] (plus possibly the sensitivity to the duration of the preceding vowel). I have zero confidence in my ability to do that, but how about you? Please let me know, because I would really like to know!