I haven’t had any personal experience, but I would send both together. It would benefit the shy clingy one to get social and it would help having their twin around

If you hold one child back you may find the younger may think you are favourint the older one …..OR the older one may resent you spending more time with the onw who stays home with you.

It is a shame they will be in the same classroom because the teacher may comapre them or may even favour one over the other. I have had no experience with twins. I do know of 2 children in a class – one is shy and the other one is sometimes a bit too boisterous and outspoken. The shy one has come out of her shell and other one has quietened down.
They have had a good effect on each other. Your shy one may benefit from starting school rather than staying shy for even longer. Does she ever stay at somebody else’s place at all ? Maybe she could have a couple of hours with another relative or close friend with children. I know a school age child who had never stayed away from home. They suddenly realised that if her Mum got sick enough to have to stay in hospital the stress would be enough without having to stay away from home for the first time. She also gained confidence and in the process of that gradually became more outgoing.

I wouldn’t hold one back as i would feel it would implicate things more
but you are their mum you know them best trust your instincts
My daughter started kindy this year she was a handful and i was worried she wouldn’t be able to sit and concentrate and i was worried for the teacher whether it would turn into a terrible problem for them trying to keep her calm, but she has flourished she is a totally different child now. She is so obedient and involved in all her class discussions and completes all her work so well! So it could help her out of her shell