My main objection to this, especially in this online world, is that anonymity means no accountability. You can publicly attack others and impugn their character, orthodoxy, whatever. You can do real damage to them. You can misrepresent truth (intentionally or not), such as Catholic doctrine. You can mislead others as to your identity (and by that, I mean more than just name). You can be a total misanthrope, be hateful, spiteful, and mean. All without any personal accountability.
How unjust is it for anonymous bloggers to tell bishops, for instance, that the bishops need to "stand up for the faith" and "have some backbone," when they don't even have the backbone to identify themselves? How much more when they make sweeping statements impugning the orthodoxy and charity of vast swaths of our bishops today?
When I'm feeling generous (especially with some of these folks I consider friends), I want to believe the stated good intentions. But when I'm feeling less so, it's hard to swallow, especially when I see examples of such vitriol from anonymous bloggers--because it's just too convenient to be anonymous online. You have nothing on the line, no accountability.
So for all those--especially purportedly Catholic (although who could verify it?)--online personalities who proclaim humility in anonymity (or whatever other personal concerns you have), I urge you to search your heart again. Is there any hint of fear that you could be held to account for what you write? Even if there's not--especially if there is not--you should reconsider because the potential evil and harm to others (through malice or inculpable error) far outweighs the potential danger to yourself, your humility or privacy. Is it not, in fact, an act of charity to do so? To take risk to yourself out of concern for others? I think our Lord would say so--"no greater love has a man than that he lay down his life for another."
If you can't take that risk and accountability, I'd humbly suggest that perhaps it would be better to not blog, comment, tweet--whatever--online at all.