Around the virtual water cooler at Language Log Plaza this morning, I asked Victor Mair about how Beijingers actually say the name of their city. I was curious, because I know from earlier experience that people from that part of China often weaken consonants in the middle of two-syllable words. For example, once in an introductory phonetics class where the topic was phonetic transcription and spectrogram reading, we worked on a phrase from a Mandarin news broadcast that included the word 比较 bi3jiao4 "rather" (as in "rather hot"). In that case, the medial 'j' was pronounced as a glide, as if the word had been written as bi3yao4. So I wondered whether the 'j' in Beijing might also sometimes be pronounced as an IPA [j].
There's a huge amount of slurring that goes on in Pekingese (Beijing colloquial), especially with regard to medial consonants of all sorts, such that the consonants beginning the second syllable of bisyllabic words are often weakened or nearly dropped altogether, and we're left with blurred consonants and sometimes what seem to be only vowel contours.
DIAN4SHI4 ("television") comes out sounding like DIAN4–I, with just the slightest trace of an SH in there (sometimes I can't hear it at all).
Even many initial consonants get blurred: WO3 BU4 ZHI1DAO4/DAO ("I don't know") comes out something like this: -O3 (B)(U)(4) (ZH)I1DAO4, with the elements in parentheses being weakened or elided altogether, especially in extremely laconic or rapid speech.
In other words, there's an enormous amount of slurring and swallowing of consonants in Beijing colloquial. A lot of times one has to guess at meanings pretty much on the basis of vowel contours.
So, to tell the truth, often when Beijingers are not enunciating conscientiously and clearly, but are just talking fast and naturally, the J of JING1 does indeed seem to get glossed over, or — as you say — becomes a glide.
Victor emphasized that fluent vernacular speech is very different from textbook speech, and what you get as a citation form from a Beijinger may be more like the textbook modern standard Mandarin version, just as a Baltimore native who normally says "balmer" may enunciate "ball tee more" when explaining to a foreigner.
In one's native language, it's often surprising to learn how much is missing, even from what sounds like perfectly clear speech. In the phonetics class that I mentioned, a native speaker of Chinese (from Beijing) transcribed bi3jiao4 as [bidʒau] and refused to believe, even after repeated careful listening, that the medial consonant was a glide, not an affricate. So I had him make a spectrogram, in which the lack of any stop gap or fricative noise was visually plain. His response was to conjecture that I had somehow hacked the computer program, in order to play a joke on him for pedagogical effect, because it was so completely obvious to his ear that the non-existent affricate was actually there.
This is an instance of the phonemic restoration effect, as a result of which the perceptions of native speakers are sometimes, paradoxically, less acute (in purely acoustic terms) than those of outsiders.
Anyhow, I think it would be interesting to take a look at the pronunciation of Beijing in some Chinese-language news broadcasts, and see how often the [dʒ] is lenited in that rather formal context.
Victor also pointed me to Bosat Man, "Backhill/Peking/Beijing", Sino-Platonic Papers #19, 1990, which traces pronunciations of the name of the Chinese capital across space and time.