They trust one another im sure, but Halo 3 implies no friendship between the two.
Whenever Arbiter tries to talk to Chief it goes like this.
"Truth is in that structure now we end this this" Arbiter
Yep, a real friendship.
It is friendship if you wanna know,but keep on mind that Chief surely never talks alot.Since Halo started we all learnt alot about him,also that he doesn't talk much.It's just his temperament and it's characteristic for him to be that ''silent'' type of person.Even if they made Halo 4 where Chief talks more than in previous games,Cortana said 10 lines and just after that he says something short and simple.Also remember how it looked between Johnson and him in Halo 2 & Halo 3.Does that mean Johnson and John were not friends?After all it's better for everyone if Arbiter n Master Chief have some kind of friendship.If not,then you should know Halo 5 could maybe bring big plot twist to whole franchise.No doubt it would impacts Halo fans too.Although my English isn't much correct hope you will find a way to understand what I'm saying.As I said before,nobody said they grew up in same neighborhood or they were playing games together to say Arbiter and Chief are BFF.