We will not spend much time on the writings of the rabbis, for there is very little of value that they can offer in this context. Some even doubt that Jesus is referred to at all: Meier writes that the rabbinic sources contain "no clear or even probable reference" to Jesus, and may be considered primarily as reactions to Christian claims. [Meie.MarJ, 96-7] Yamauchi cites Twelftree (see also Twel.GosP5) as saying that the Talmud references are "of almost no value to the historian in his search for the historical Jesus," although a contrary view is also offered by Wilcox, who recommends that the material may be used, albeit cautiously. [Wilk.JUF, 211]
In light of the above, there is no need for us to run over each of the rabbinic citations here. The single point that may be derived from them is, again, that it provides no indication that Jesus was a mythical figure; inasmuch as it accepts Jesus' historicity, and does not doubt it, it provides positive evidence that Jesus did exist.
Wilson [Wils.EvJ, 65] agrees with this assessment, saying that "From the fact that (the rabbis) concentrated on smearing (Jesus') legitimacy (rather than focusing on the issue of Jesus' existence), we may deduce that they had no grounds whatever for doubting his historical existence." And France agrees that, "Such polemic, often using 'facts' quite distinct from what Christians believed, is hardly likely to have arisen within less than a century around a non-existent figure." [Franc.EvJ, 39]
Some scholars do argue that the references also have value in what else they affirm, do not deny, and provide no contradictory tradition for: That Jesus had disciples, worked miracles, and was executed on the eve of Passover. (See particularly Harv.JesC, 30-1.) Aside from that, the Talmud and other Jewish references are of marginal value.
The Talmud authorities only had one source of information about Jesus: Christians. Being that much of the alleged references are polemical responses to Christian claims, there is no indication that the data was arrived at independently.
Actually, we have no clue about the souces for the Talmud - much less do we know that (1) the Talmudists had only one source; and (2) that is was Christian! There is not the slightest evidence to support this presumption [also contra ChilEv.Stud, 444] We simply do not have enough data to say one way or the other.
Further, we may point out again, as Wilson has, that if there were any hint that Jesus was a mythical figure, we would expect that the Talmuds would aim some polemics in that fertile direction. As it is, there are no such statements; and it strains credulity to say that the authors of the Talmud would have simply taken Christians' word for Jesus' existence if evidence existed to the contrary (and it would have existed, had that truly been the case -- there would be "holes" in the historical record big enough to drive a Borg cube through).
The anger and distaste expressed in the Talmud for Christianity leads to a solid inference that any useful information against it would have been taken up as a weapon. Therefore, they may be taken as an independent and reliable witness for the mere fact of Jesus' existence, while not necessarily that for actions and sayings of Jesus. And of course, just because they are polemic, this does not automatically mean that they are not independent.