It looks like by default JIRA swallows runtime exceptions, i.e. I don't see them in the logs. Is there a way to enable logging them?
As a rule JIRA does not swallow runtime exceptions. The stuff which gets logged depends on where the problem occurs.
For built-in stuff most of exceptions are logged if they are unexpected (to control it configure logging via atlassian-jira/WEB-INF/classes/log4j.properties) and Logging & Profiling screen (settings there are lost upon JIRA restart). Try setting some loggers to DEBUG level and see if you then see the exceptions.
For 3rd party plugins - everything depends on how plugins handle runtime exceptions and whether they decide to log it and using which level (from DEBUG to FATAL).