You've got people here telling you that your kids aren't ready to train. I say, YOU know your kids. They do not.
I practiced EC (elimination communication) with our daughter since was 5 months old. Yes, it is infant potty training, but it is low pressure and it is more about ME watching HER signals and not trying to get a 5 month old to tell you they have to go potty. All babies give signals when they have to go. They have an instinct to not soil themselves, we teach them to go in diapers and then expect them to unlearn it later.
That being said, I really felt at 18 months my daughter was to a point where she was ready to go full on potty training. She could take her pants off, she KNEW when she needed to go and often times would tell me she needed to go we would go to the toilet and she would actually pee. So, I ditched the diapers (except for nap and bedtime).
I will say, pull ups are evil. Yes, they save a lot of laundry, but I really think kids GET that they are just a glorified diaper. So, I only use pull ups at nap time, or if we are going to be gone for a long period of time and I think we might have issues. (but quite honestly, whenever we are out she hasn't had any accidents)
It seems like your kids are aware AFTER they go. You need to have them tune into their bodies urges BEFORE they go. A good way is to let them run naked. Or put them in a pair of undies... when you see they are going say 'You're going PEE!' They will get that body awareness.
You can also accomplish this by giving them lots of water and then taking them to the toilet every 10 minutes or so. I find with my daughter sometimes she needs help relaxing and letting it go. So, we take a small toy or book. Most often a book. Reading the book helps her relax and then she can pee. Yes, sometimes we are there for 10 minutes...but she is learning to control that muscle. The muscle that we use to go pee is always contracted...so we have to consciously release it and relax it. It takes some time for kids to get that and master it. Patience is KEY! I prefer panties because she doesn't like the feeling of the wet panties and pants. It's more difficult for her to tell with pull ups.
We also do a sticker chart. Morning and afternoon opportunity for a sticker. No accidents... she gets a sticker.
My daughter is 20.5 months old. We've been doing this since she was 18 months. Do we have accident free days? Yes. Not every day... but we do have several. I think at 2 years it will be easier for you.
Patience, persistence...those are key. Also, I know I have to take her to the potty regularly. I know approximately how often she has to go...and I take her to the potty regularly. The more she goes in the potty, the more she learns that is what we do.
I started with a little child potty chair... but soon she wanted to sit on the big toilet like mama. So, I got a seat to go on the toilet. A friend of mine bought this thing called The Potty Stool through Amazon. It's AWESOME. At 18 months of age she could climb up and put herself on the potty. She loved that. She is very independent and loved having something else she could do on her own.
Good luck. If YOU think your kids are ready, then I'm sure they are. Don't let someone else who is not their mommy tell you that you don't know your kids. Mommy instinct... it's powerful.