Many religious theists would like atheists to be more "moderate" in their criticisms of religion, religious beliefs, and theism. They contend that atheists are too intolerant of religion and should show more respect to religious beliefs. There are good reasons to dispute how religious theists use the terms "tolerance" and "respect" in these contexts and one way this can be demonstrated is by asking just what sort of "moderation" they expect atheists to demonstrate.
Is a moderate atheist a person who disbelieves in gods only sometimes rather than all the time? This would mean that they aren't always rejecting the idea that gods and religion are central to human existence and sometimes agree with theists.
Is a moderate atheist someone who doesn't mind if people believe things they consider silly, dangerous, irrational, ill-founded, etc.? This would allow people with those beliefs to continue promoting them without fear of contradiction, challenge, questions, or dissent.
Is an atheist "good" when they don't bring up uncomfortable questions about why allegedly "peaceful" religions have been involved with large-scale wars and brutality throughout human history? This would allow religious believers to continue acting as though their religions were unimpeachable forces for good, for social order, and for morality.
Is an atheist "acceptable" if they are willing to indulge Christian anti-Semitism, Sunnis killing Shiites, Hindus attacking Muslims, and other forms of inter-religious violence? This would allow religious believers to continue bashing and killing each other until one stands tall atop a pile of dead bodies.
Is an atheist "tolerant" if they don't mind when religions teach that women are inferior to men, that women can be killed if they violate the "honor" of families, that religions and religious beliefs should be granted special privileges, that gays should be treated as disordered and second-class citizens, that women shouldn't be allowed to control their own bodies, and so forth? This would allow extremists among religious believers to exercise greater control over society and everyone in it.
None of these really make much sense, but when religious theists say that atheists are too extreme, it's usually in the context of atheists criticizing beliefs they consider false, irrational, silly, or even harmful. If it's OK for atheists to continue criticizing all of these beliefs and behaviors described here, then what could it possibly mean for atheists to be more "moderate"? If it isn't OK for atheists to continue criticizing all of these beliefs and behaviors, then why not — and when will religious theists cease criticizing atheism, godlessness, and not being religious?
Religious believers are obviously entitled to their beliefs and they should not be persecuted for them. They are not entitled to be free of criticism for those beliefs nor do they have a right to have beliefs which are not criticized. It seems to me that demands for atheists to be more "moderate" are ultimately demands for atheists to self-censor and not bring up uncomfortable arguments, facts, and ideas which conflict with religion's long hold on people's minds.
This is a completely unacceptable demand. If a person generally keeps their opinions to themselves and doesn't actively promote them in public, then it would be unfair, unreasonable, and rude for others to keep pushing this person and trying to criticize what they think. People do have some right to be left alone with their beliefs when they are leaving others alone with their beliefs.
Once a belief, idea, opinion, or principle is put out in public and promoted as a just, good, or even superior position to adopt, then any expectation of being left alone must be abandoned. Personal attacks are generally out of line (excepting of course extreme positions such as those advocating violence, discrimination, etc.), but even very strongly worded criticism of the opinions themselves must be accepted as part of the marketplace of ideas.
