I had this with my twins when they started standing, a bit older for them as they were prem but from what I understand it’s a good sign from a developmental perspective - but good God is it tiring and stressful. There is a sleep regression at 6 months from what I remember reading and my HV said usually when a new skill is gained or if your DC is approaching a new developmental leap, their sleep can become quite interrupted.
Also at 6 months they might have their first sensations of understanding you and they are in fact, separate beings and this can lead to a phase of separation anxiety.
I had to do sleep training with mine for my own sanity. Not crying it out though.
Firstly in their own room, this will set you back a bit whilst they get used to being in their own room. Introducing them to it in a way that makes it fun and comfortable for them so they get familiar with the room in a positive way.
Then, if little one whines, don’t go in, they may just have a bit of a whine and then settle back down - DS did this as he was standing automatically with no conscious decision to do so whilst asleep and the motion woke him up. He’d make a few little noises of general grumpiness and then settle back down. On occasions where he immediately started crying I went in straight away, picked him up and cuddled, but didn’t move away from his cot (unless he needed a nappy change of course) and then placed him back, this would cause him to get upset again so then I would place a hand in with him, resting on his stomach so he knew I was still there. Then after 10 or so minutes I’d tell him “time for sleep now” and leave the room. Stand outside with the door closed and wait for if he would settle.
Now sometimes he’d shout out when I was leaving the room but unless it was full on wailing I would resist the urge to go back to him and stand outside of the bedroom door.
I’d repeat this as many times as necessary.
DD is a bit different, she has always been such a chatter box so would stand babbling away rather than crying. In those instances I’d only go in if she was loud enough that it might wake her brother or if it became clear she was working herself up.
There are lots of good sleep training methods. Some prefer the ‘vanishing chair’ approach, others might sleep in the babies room with them whilst they get them used to sleeping there.
Also do you have a set routine in place for bedtimes? At 9mo I had to adjust mine to remove bathing (DD hates the bath so in no way soothing for her). I paired it down to nappy change, bottle (in as dark a room as I could manage to see in), sing “you are my sunshine” placing them back into sleeping position several times throughout and then I say goodnight to both and leave the room.
Good luck OP and I hope you find a method that works for you! All I can say is you have to stick to whatever method you choose as best you can, it can take a while for it to embed and there might be times you’re so knackered that you want to rush through the bedtime routine, but stick to it as best you can and 
PS Tell your DH to screw off unless he wants to take the front line on sorting your DC for bedtime, if he’s anything like mine was at 6mo, plenty of unhelpful suggestions but little actual input in trying to sort it