It is totally up to you. They don't "need" a routine. I started putting DS1 to bed before us in the evenings when he started crawling (8 months), because otherwise I'd spend my evenings chasing after him which sort of defeated the point.
DS2 was clearly bothered by light and sound in the living room during the evening from about 3 months so we started putting him to bed on his own then. They are all different and IME you'll know when it feels right.
I get annoyed by the whole narrative about sleep TBH, hate all the insistence on routines and "training", fine to give advice if that's what somebody wants/has asked for but it's a choice, not a necessity. The biological norm (ie what happens if you do nothing) is that they don't sleep through or independently until they're about two years old. Honestly, if you don't like the sound of that, that's absolutely fine and understandable, but some of us prefer to manage it by co-sleeping instead and that's fine too. (And some babies don't respond to routines and sleep training, so their parents should be reassured that it's not something they are doing wrong, either.)
There is not very good evidence for the being in the same room for all sleeps thing. It might have a slight protective effect, but it seems to be the most important for the long stretch of night time sleep. The UK is the only country to include it in advice from anywhere; it isn't even part of NHS advice, only Lullaby Trust. Personally I'd probably disregard it even for that sleep by 4 months though as the overall risk of SIDS has dropped. Although as said, I prefer to bedshare as I find it gets me and DS2 the most sleep and fits with my overall approach.