Abby Wood and Doug Spencer have posted this draft on SSRN.  her is the abstract:

In recent years, the courts have deregulated many areas of campaign finance while simultaneously upholding campaign finance disclosure requirements. Opponents of disclosure claim that it chills speech and deters political participation. We leverage state contribution data and find that the speech-chilling effects of disclosure are negligible. On average, donors to sate-level campaigns are no less likely to contribute in subsequent elections in states that increase the public visibility of campaign contributions, relative to donors in states that do not change their disclosure laws or practices over the same time period – estimates are indistinguishable from zero and confidence intervals are narrow around zero. Moreover, we do not not observe heterogeneous effects for small donors or ideological outliers, despite an assumption in First Amendment jurisprudence that these donors are disproportionately affected by campaign finance regulation. In short, the argument that disclosure aggressively chills speech is not supported by our data.