Why doesn't Vermouth tell Gin/Black Organization that Kudo Shinichi is alive?

[Jessicaz140]

Mmmm...

[kyuuketsuki]

She feels indebted to him for saving her life twice.

Also, my own personal theory is that she wants the black organization to be brought down. Why else would she say "we only need one silver bullet" or something to that degree...

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[Stopwatch]

There are various reasons and I doubt we'll ever find them all out but I reckon these are some of them:
1) Vermouth knew Yukiko, it's likely that she was influenced by Yukiko's easy going and accepting personality and still thinks of her as a friend even now.
2) The entire 'saving Vermouth's life' thing that's already been mentioned, though it can't just be that considering that Vermouth would have 'payed back her debt' at this point because of the amount of times she could have killed/given away Conan. I personally reckon that it was either that she was at a low point in her life (the entire 'there is no God thing' she had going) and what Shinichi and Ran said when they saved her had a huge impact (the entire 'Angel' thing shows she may think it's divine intervention or at the very least she's very, very thankful) or a combination of both.
3) She has a soft spot for children and teenagers: this has been shown plenty of times (the fact she tends to let any children who may have seen things to do with the Org alone because 'no-one will believe them' when they could in the future pose a serious threat for the Organisation (Jodie, for example) and that when she was disguised as Araide teenagers at Ran's school commented that s/he'd become even nicer to them during that time). This could possibly be due to some unknown part of her past or just that she likes how they are 'naive'.
4) Also, she probably sees Shinichi as her chance to destroy the Org and escape from it, as mentioned by kyuuketsuki.
---Those are the reasons I can come up with off the top of my head at least  :D

[Shuusgirl]

There are various reasons and I doubt we'll ever find them all out but I reckon these are some of them:
1) Vermouth knew Yukiko, it's likely that she was influenced by Yukiko's easy going and accepting personality and still thinks of her as a friend even now.
2) The entire 'saving Vermouth's life' thing that's already been mentioned, though it can't just be that considering that Vermouth would have 'payed back her debt' at this point because of the amount of times she could have killed/given away Conan. I personally reckon that it was either that she was at a low point in her life (the entire 'there is no God thing' she had going) and what Shinichi and Ran said when they saved her had a huge impact (the entire 'Angel' thing shows she may think it's divine intervention or at the very least she's very, very thankful) or a combination of both.
3) She has a soft spot for children and teenagers: this has been shown plenty of times (the fact she tends to let any children who may have seen things to do with the Org alone because 'no-one will believe them' when they could in the future pose a serious threat for the Organisation (Jodie, for example) and that when she was disguised as Araide teenagers at Ran's school commented that s/he'd become even nicer to them during that time). This could possibly be due to some unknown part of her past or just that she likes how they are 'naive'.
4) Also, she probably sees Shinichi as her chance to destroy the Org and escape from it, as mentioned by kyuuketsuki.
---Those are the reasons I can come up with off the top of my head at least  :D

Agreed.  It's probably all these reasons.  While I don't think that Sharon and Yukiko were super close (meaning they didn't meet up whenever they could, etc) because then Shinichi could have recognized her, I think that Yukiko's cheerfulness and kindness touched her. 
As for number three, yeah, that's been proven over and over to the point where I'm wondering if we're going to get some more backstory here...I hope so!!!  Although I wonder if it's not so much cause they're "naive" as much as innocent. 

[Kaia]

"A secret makes a woman woman."

I think this is also a reason.

[JohnTitor]

She has a crush on him.

[skdc8595]

She wants him to destroy the organization.....

And owe her life to him...