The group i'm part of has run a number of DH games over the years and they've all worked out well.
As basic acolytes you have no idea what political leanings your Inquisitor has.
As trusted acolytes you begin to have an idea, based on the sorts of direct orders you get issued (don't harm the xenos-witch, burn the room the daemon was summoned into with promethium etc), as to the leanings of your Inquisitor . Of course without Forbidden Lore: Inquisition and some greater idea of the political game Inquisitors play (such as being aware of infighting and factionalism) this won't amount to much.
Finally as inner circle, that is your Inquisitor's most trusted acolytes, you have a pretty good idea what is going on. Words such as Horusian or Xanthite may not be bandied about casually, but its pretty obvious if your master is constantly in contact with xenos, or is a powerful psyker that utilises forbidden rituals.
I think it isn't really until Ascension level that players should really be concerning themselves with the factions or aligning themselves with one in particular. Sure, you can generally play a puritanical Emperor worshiper who burns everything even slightly deviant in sight. And sure, you can also play a lose canon who is constantly reprimanded and given punishments by their superiors for not utilising forbidden items and techniques and consorting with mutants, heretics and xenos filth. But to really divide Puritan and Radical up into their factions and put specific names to them and allow the players to align with one, i think it's really high powered, end game territory.
We had a long running game (went on for just over a year with a 5 hour session every week) in which our group of roughly 6 players (it fluctuated from 5 to 8 in reality, with characters killed off and new ones rolled up) followed the escapades of their clearly unusual Inquisitor. Seemingly just quirky at first, as we progressed in trust and status it became clear that our master had contacts in multiple Ordos of the Inquisition, utilised heavily forbidden things such as cloning technology, and consorted with daemons. Now the way this was handled was very good. As the players had been allowed to put their own spin on how their characters acted for the first 1/3rd of the game, when we entered the second 1/3rd it became clear that the group was split into radicals, some puritans and some neutral characters. The radicals, spear-headed by my sorcerous adept (he had funkier psychic options that the actual psyker!), formed their own clique within the acolyte cell and received additional instruction from our Inquisitor.
Meanwhile the few puritans we had were given similar instructions to their clique and while it became apparent that we were being set off against one another, with only the neutral characters to keep us in check, it was apparent that the Puritans were being distracted whilst the Radicals did the real work. This all led to some great confrontational moments, daemonic incursions and large firefights.
Ultimately the game ended with a huge deus ex machina, with Scintilla being engulfed in an artificial warp storm or something like that… it went a bit crazy. And while all the characters were essentially killed off in an "unhappy ending", our actions and alleigances during the game determined some of our fate.
Obviously not all games will be as long running or convoluted as ours, and since we played there have been multiple source books with extra cell packages that can help bind a group cohesively to a common cause, but i think it is important to lay down some ground rules before play as the GM, such as saying 'players can do anything they want, but it's an Ordo Xenos game' or 'Puritan only, but up to you how'. Conversely the players should have an idea of how they'd like to progress their characters and so what options they'd like open in the game.
Sorry for such a long winded response but i do feel it's not a simple question to answer and some insight into previous handlings of it may be enlightening for some.