The bill supersedes privacy and communication laws, but is (a) opt-in and (b) severely limited in scope.
Specifically: CISPA provides a positive authority for sharing only "cyber threat information", which is defined in the bill: (i) information about a vulnerability, (ii) information about a confidentiality/integrity/availability threat, (iii) information about denial of service or destructive attacks, and (iv) efforts to hack into systems and exfiltrate data.
The bill incudes language that explicitly exempts the kind of stuff Aaron Swartz got caught up into: it exempts attacks that "solely involve violations of consumer terms of service or consumer licensing agreements and do not otherwise constitute unauthorized access.". That exclusion is repeated multiple times in the definitions section of the bill.
The bill explicitly does not cover individuals, in a fashion that the bill's authors say affirmatively prevents it from being used to allow ISPs to share individual customer records.
So: back to you. What specific state or Federal privacy measure is compromised by CISPA, and how?