Does access to logcat need root? My device is already rooted, this is why I'm asking.

Do you need root to monitor the logcat stream on the phone? If I ran the logcat command from within a Terminal on the phone, would that work?
These free apps in the Market claim to be able to display your logcat, and don't make any mention of root. Should only take you a few minutes to try them out:


CatLog
aLogcat
LogViewer


EDIT: Thanks to @mente for reminding me that these apps will only let you view the full logfile in older versions of Android. Android 4.1 brought in a new security model around logfiles that meant that apps could only read their own logfiles and logfile entries, and can't read logfile entries written by the system or other apps.

There is however still both a way for an app's user to generate a logfile of the app's recent log entries to be sent to a developer, and for a phone's owner to grant permissions to allow an app to read the logfile without full root, but this does require you to use an ADB command, which unfortunately takes it out of the reach of most normal phone users:


  If you want the user to give you debugging information, you can have
  them generate a bug report with power + volume down + volume up which
  includes the logs and lots of other data, and automatically brings up
  their e-mail app to sent it all (plus a screenshot).  We were just
  discussing that we should have an easier way to generate these as
  well, I am going to look at adding something to the settings app.
  
  I also have started introducing the concept of a "development"
  permission, which read logs is classified as.  This allows the app to
  request the permission, but not get it at install.  You can however
  grant it with an adb shell command once it is installed.  At some
  point later I expect to have a UI in the system for doing this, but we
  are going to hold off on that to be careful about how we present this.


Excerpt from a post on Google Groups by Dianne Hackborn, Android Framework Engineer. My emphasis.

The mentioned ADB command to grant an application permission to read the logs is:

adb shell pm grant &lt;pkg&gt; android.permission.READ_LOGS


This setting should survive reboots and upgrades of the app (but not uninstalls/reinstalls).