I intimately understand both of your priorities and predicaments.
You see each other’s view and conclusions as the greatest threats to the way you want the world to be, the explanations you find satisfactory, the hierarchy and level of societal control that you deem appropriate, and the rights you deem you deserve. Atheists are often very tired of being culturally held ransom for their religious dissent so they come out blazing against theistic hegemony. Theists often feel that loud atheists are encroaching steadily on their religious rights. Conversations between the two despite the best intentions steer from a civil introduction towards a brutal pub brawl whether alcohol is involved or not.
There are a few things I’ve discovered after being a core and committed member of both groups at different points in my life, that I think can build bridges and tunnels towards better, more honest and fruitful conversations between each other.
1) Theists and atheists are both equally human.
I know this seems almost condescending and certainly mundane, but in the midst of our tribalistic proclivity to draw lines around our distinctives and bolster solidarity and identity within our tight-knit communities, we invariably lose sight of the other’s humanity. Consciously or not we downgrade those with whom we disagree as a means to feel better about ourselves and to justify treatment the other with disrespect. Because of our equal humanity, we both share equal human rights, protections, and opportunities under the law.
2) Both Theists and atheists don't believe in god or god's.
I understand that this sounds almost ridiculous and offensive to the theist but permit me to explain. As a theist, there are literally tens of thousands of gods that you neither acknowledge, believe in, nor worship and would find it ridiculous for anyone to demand justification for why you don’t have an altar for them, or worse be demanded to have your freedoms dictated by their religious decrees. This perhaps can provide a meeting place, common sacred ground if you will that might change the attitude you have towards conversations about contentious religious matters and give you both more empathy for each other.
3) Both theists and atheists can make poor moral choices.
Neither the theist nor the atheist has the moral high ground based on their beliefs but demonstrates their morality based on their actions; especially when no one is looking. That being said, it is often a stereotype in primarily religious societies that theists are presumed virtuous while atheists are deemed immoral from the outset. Let's stop perpetuating these myths about each other as they poison our impressions and expectations of each other
4) All theists and atheists can be susceptible to bad arguments, failures in logic and critical thinking or unchecked biases.
Atheists often imagine themselves as more rational or unbiased than theists but this isn’t true. Attacking someone’s intellectual capacity based on their religious convictions is disingenuous and uncharitable. Self assigning superior intelligence just on the grounds of one’s atheism is a delusion of grandeur. Unless we have good reason to think someone has an intellectual impairment, it more charitable to assume that any apparent lack of cohesiveness to one's arguments is due to a genuine misunderstanding that can be rectified through pointing that out with respect. Insulting another’s intelligence is not a way to prove the validity of your own argument and gets us nowhere near a fruitful outcome.
5) Some theists and atheists have much more in common than what they disagree on.
Start with the common ground as a way to understand well what you agree on as a bridge to better understand the what and the why of your disagreements. You may be surprised by some of the joy you could unearth in your mutual interests.
6) Freedom to hold and express one's religious/irreligious convictions does not give one a right to enforce those on anyone else.
You are not free to force anyone to accept your religious views nor live by the moral codes of those convictions. In a democratic society, we are all contractually subjected to laws which govern conduct, responsibilities, rights, and freedoms of all of its citizens regardless of their religious convictions. When our rights and freedoms clash and collide we must certainly need to figure out a way to dialogue with each other to find a point of reasonable compromise without a denial of dignity.
An atheist and a theist meet at a pub to talk over 2 beers, What happens next?