I'm testing my managed package (it's Beta now) trying to install it to new Dev Org. It's weird thing but seems like debug statements inside my package are not shown. Is that known bug/issue or I need to configure/enable something to show debug statements.
Debug logs raised from within a managed package are not exposed within the installing Org by default (Since Winter '12). This includes beta managed packages and full releases. See Logging from a deployed managed package.
You now need to use the LMA (License Managing Application) with the managed package to log into the Org and see the debug logs. See Subscriber Support Console and Troubleshooting in Subscriber Organizations
My experience with betas is that they usually aren't in the LMA yet as they are still in the early stages of development. I.e. they haven't been reviewed yet so can't be in the LMA.
As Keith suggests in the comments, if you have paid support you can create a case to have the debugging exposed (include your namespace and packaging org id plus a timespan for how long the logging should be exposed.). I've had zero luck with support without paid support (even for reporting platform bugs and Org specific issues). They just close cases and direct you to the developer forums. The developer forums aren't going to be much help here, as support need to access the Org to change the logging settings.
I raised the idea Expose logging from a Beta managed package in the deployed org a couple of years ago. Your vote would be appreciated. I'd really like a checkbox when making the beta managed package that indicates if I want logging exposed. Maybe a dropbox logging level to expose. I.e. Errors but not debug.
You could go so far as to implement your own logging. This is potentially expensive in terms of records and database transactions, but it will work with betas and full releases alike. You could use a hierarchy custom setting to control what is logged and for whom. You might want to consider rollback behaviours when dealing with exceptions.