Many people all over the internet have written their thoughts on how this pairing could actually work. Some have even given their thoughts on how it would never work. I respect the opinions of everyone and I do not expect you to agree with me. Though if you are reading this, I hope I do not bore you.
First off I will start with Izaya. Many people have written fanfictions where he's had past relationships, slept with clients or is just simply labelled as a "slut". My first point on this has got to be that I highly doubt Izaya would do any of this. I think Izaya is a virgin seeing as he thinks of himself as a god above humanity. He would not give himself to anyone; he probably thinks sex is just an unnecessary need and it would make him seem human if he gave into it. As for relationships, he doesn't let anyone get to know him well enough to even get to that stage, not to mention he himself has said he cannot favour any one human over the others. His only real friend is Shinra, and even then they are not normal friends. Celty only works for him, just like Namie does. They do not like him, just tolerate him. He says many times that he loves all humans equally, but he does not see Shizuo as a human.
As for Shizuo I think he would also be a virgin and not have any experience with relationships. He wants to get along with people and probably wishes he could one day actually be in a meaningful relationship with somebody. Most are afraid of him for his strength and he is afraid of hurting people he cares about with it. The only people he really has to talk to are Celty, Shinra, Kasuka and on occasion, Simon. He has enough 'friends' to be content, but I don't think he is anywhere near happy. Apart from his 'friends' and his brother, Izaya is the only other person not afraid of him.
It is no secret that Izaya and Shizuo have fought since high school. No sooner had Shinra introduced them, Shizuo disliked Izaya. I cannot even begin to imagine the amount of surprise felt by Shizuo when Izaya cut him. Izaya had seen what Shizuo is capable of and was not scared at all. Shizuo would obviously fight back rather than think. This became a regular routine for them and since then they would fight every day. The city of Ikebukuro knows their names, faces and fights. They are easily the two most dangerous men in Ikebukuro, Shizuo for his strength and temper, Izaya for his manipulative personality and cunning.
Now, I am not one of those fangirls who say they love each other, the fights are faked and what have you. I have two ideas on the matter. The first is that Izaya has come to realise he does care for Shizuo, yet he will not admit it to himself let alone anyone else. Shizuo could also like Izaya, but not realise it. Izaya would instigate these fights with Shizuo as it's the only way of spending time with him; Shizuo just fights back without thinking about why Izaya keeps coming back. My second idea is that they both care for each other but are too engrossed in the idea of hating each other to see it. I think they both need the other. Shizuo is the only unpredictable human Izaya has come across, thus he was labelled as a monster to separate him from the humans that Izaya loves so dearly. For Shizuo, Izaya is one of few people who do not fear him and he also does not need to worry about harming Izaya.
I believe that Shizaya (or to a lesser degree, Izuo) could work. I am not saying it will happen or it needs to, but I think it could. Neither are afraid of the other nor do not worry about harming them. They state they hate each other, yet Izaya can't go long without pestering Shizuo and Shizuo cannot help but chase Izaya. I think if they stopped fighting and actually talked or spent time together, they could get along. I believe if that happened they would become close friends at least, sure they would bicker and argue, maybe even fight sometimes, but they would still get along.