Let's break it down by category.
What information does Carrier IQ monitor? Trevor Eckhart says (depending on the phone manufacturer) it receives each key pressed/tapped, the location of any tap on the screen, the contents of all text messages received, the name of each app that you open or switch focus to, information about each call you receive, your location each time location is updated/queried, the URL of each web page visited (including URL parameters; yes, even for https URLs), and possibly other information about each HTTP request. I have not seen anyone dispute these claims.
Note that this is information that is monitored by the Carrier IQ application; that doesn't necessarily mean that the application does anything with the data, stores it, or allows it to leave your phone.
What information does Carrier IQ record on your phone? It is hard to get clear information on what information might be stored in your phone on persistent storage or log files. Does Carrier IQ log the information that it receives? I don't know.
Carrier IQ says that their software "does not record, store or transmit the contents of SMS messages, email, photographs, audio or video", and they have said "we're not storing" keystrokes and that they "do not record text messages". However, they also say that they do "record where you were when [a] call [is] dropped, and the location of the tower being used". Lookout says "it doesn't appear that they are sending your keystrokes straight to the carriers". Dan Rosenberg seems to suggest that the Carrier IQ application is "recording events like keystrokes and HTTPS URLs to a debugging buffer", but it is not clear to me where that debugging buffer is stored (just in the memory of the Carrier IQ application? or on persistent storage of some sort?), and it is always possible I have misinterpreted his statement or read too much into a brief phrase. Dan Rosenburg subsequently elaborated, finding that on one particular phone, CarrierIQ can record URLs visited (including for HTTPS), GPS location data, and phone numbers, but not all keystrokes, not the contents of SMS texts, and not the contents of web pages browsed.
CarrierIQ has subsequently clarified that their software does record "the telephone numbers the SMSs are from and to".
Trevor Eckhart said that the Carrier IQ software on his HTC phone recorded a lot of personal data (keys pressed, SMS texts, etc.) into a debugging log file, so this information is stored in the clear on his phone. Carrier IQ has subsequently confirmed this finding. Carrier IQ says this is because the debug capabilities remained switched on; it sounds like they are blaming HTC for not deleting or disabling the debugging code in the Carrier IQ software. It is not known whether a similar problem may be present on phones from other manufacturers, or if this is limited to just HTC phones.
What information is transmitted to carriers? Carrier IQ says that only diagnostics information and other statistics leave your phone: "For example, we understand whether an SMS was sent accurately, but do not record or transmit the content of the SMS. We know which applications are draining your battery, but do not capture the screen." Dan Rosenburg says that the software can also report your location (GPS) in some situations. Carrier IQ has confirmed that their software captures phone numbers dialed and received and all URLs visited, if enabled by the carrier.
However, Carrier IQ also says that the amount of information that is sent to carriers is up to the carrier, and agrees that the Carrier IQ application has the capability to transmit what applications are being used and what URLs the user visits. Some of the carriers have not been very forthcoming: e.g., Sprint says they "collect enough information to understand the customer experience with devices on our network and how to address any connection problems, but we do not and cannot look at the contents of messages, photos, videos, etc., using this tool" (not very specific); AT&T says their use of Carrier IQ complies with their published privacy policies, but hasn't said anything more. Other carriers have been more explicit: Verizon and RIM say they don't use Carrier IQ and they don't pre-install it on any of their phones. Apparently T-Mobile uses Carrier IQ, but I have not yet found a statement from them.
Carrier IQ has subsequently disclosed a bug in their code which may cause it, under certain special circumstances, to capture the content of text messages and inadvertently transmit it to the carrier, as the result of an unintended bug in their code.
How is the information transmitted to carriers? Carrier IQ says says that any information that is transmitted off the phone is sent over an encrypted channel to the carrier. I haven't seen anyone dispute this statement.
Can carriers or others command the application to change any of this? I don't know. I can't tell if there is a way that carriers or Carrier IQ can send a command to the Carrier IQ application to cause it to collect, record, or communicate more information than it does in its normal operating mode.
Trevor Eckhart says that carriers can "push" a data collection profile to a phone. He also says that the profile specifies what data is collected, stored, and transmitted off the phone by the Carrier IQ application, and that any data that is received by the Carrier IQ application is potentially eligible to be transferred off the device, if the profile specifies that. He suggests that a "portal administrator" (at the carrier, presumably) thus has the ability to target a particular subscriber, push to them a profile that causes the phone to transmit a broad variety of information (keys pressed, contents of text messages, URLs, etc.) off the phone, and then can view this information. If this is accurate, it suggests that, even if the application does not normally transmit this information off the phone, the carrier has the ability to force the application to do so. It is not clear if there is any notification to the user or any attempt to gain consent before this occurs. I have not seen any independent analysis of these claims.
CarrierIQ has subsequently confirmed that it is possible to send control messages to the CarrierIQ software via SMS, to command the CarrierIQ software to perform certain tasks. CarrierIQ has not clarified what is the full range of commands that can be sent, or how the CarrierIQ software authenticates these command SMSs to make sure they are not exploited by attackers, so it is difficult to assess the risks associated with this feature.
Other information sources. Wikipedia has a page on Carrier IQ, which includes some updates, a list of carriers and handset manufacturers who do or don't deploy Carrier IQ, some reactions from policymakers, and lawsuits against Carrier IQ.