Perfect happiness, even in memory, is not common; and there were two points on which she was not quite easy. She doubted whether she had not transgressed the duty of woman by woman, in betraying her suspicions of Jane Fairfax's feelings to Frank Churchill. It was hardly right; but it had been so strong an idea, that it would escape her, and his submission to all that she told, was a compliment to her penetration which made it difficult for her to be quite certain that she ought to have held her tongue. (Chapter 27)

Emma’s conscious bothers her after the Cole’s party. She has a feeling she should not have been openly airing her suspicions about Jane Fairfax to Frank—she thinks she may have transgressed the duty of woman by woman. I agree and believe she did. Even if she had evidence of her suspicions, which she does not IMO; it is on the ugly side of gossip and just perhaps the first lady of Highbury should be above it. I can see that Frank’s submission (egging her on really) is a boon to her vanity but I think the idea that his flattery creates a situation where Emma was right to tell him her suspicions is a rationalization she creates to sooth her conscious.

…till Jane Fairfax was quite ready to sit down to the pianoforté again. That she was not immediately ready, Emma did suspect to arise from the state of her nerves; she had not yet possessed the instrument long enough to touch it without emotion; she must reason herself into the power of performance; and Emma could not but pity such feelings, whatever their origin, and could not but resolve never to expose them to her neighbour again.

…Emma wished he would be less pointed, yet could not help being amused; and when on glancing her eye towards Jane Fairfax she caught the remains of a smile, when she saw that with all the deep blush of consciousness, there had been a smile of secret delight, she had less scruple in the amusement, and much less compunction with respect to her. This amiable, upright, perfect Jane Fairfax was apparently cherishing very reprehensible feelings. (Chapter 28)

In Chapter 28 first Emma feels pity for Jane and her powerful emotions but then cannot help but be amused at Frank’s pointed questioning of Jane which is obviously making her uncomfortable. Jane smiles and Emma perceives it to be a smile of secret delight at her reprehensible feelings for Mr. Dixon. Emma then feels at liberty to have less scruple for being amused at Frank’s insinuating questions to Jane. Emma wishes he would not be so pointed and tells him so. I think Emma is pleased to have yet another reason to think Jane less than upright and when Franks says Jane ought to feel shame if she has done wrong Emma agrees, I think transgressing the duty of woman by woman yet again. It seems to me Emma cherishes some reprehensible feelings herself in that she enjoys her ideas of Jane’s fall from grace too much. Is there really a rivalry between the two women or is it all a product of Emma’s jealousy.