I would personally like to see this as a service from someone else other than the CA.
One of the exciting things about Certificate Transparency is how it can help in the case when a CA has been hacked, when it has a malicious insider, or when a government tries to force it to issue a fraudulent certificate. In these cases the CA might not be prepared to tell people about the problem, but CT can still reveal what has happened.
For these cases, someone independent from the CA would probably be best-positioned to notice and publicize the problem, because the independent party will presumably not be subject to the same problems that facilitated the misissuance in the first place.
I would also agree that CAs should not consider other CAs’ issuance likely to be malicious, as the web PKI explicitly allows certificates from several different CAs with overlapping subject coverage and validity to coexist, and we’ve seen a number of cases where people do this intentionally. For example, I saw one site where a web developer got a Let’s Encrypt cert for testing purposes while initially rolling out HTTPS, and then switched to a different paid certificate for the public deployment. We’ve also seen overlapping validity when people use a CDN like CloudFlare, where they may have a Let’s Encrypt cert for their origin server and a different cert for the CDN service. Because of cases like this I don’t think we can make a presumption that overlapping issuance is malicious or even suspicious.
But, I agree that it would be valuable for more domain registrants and sysadmins to be able to get notifications about all certificate issuance for their domains. Most of the time, I think conflicts of interest and single points of failure can probably best be minimized by having these notifications come from someone other than a CA.