Share Button coming in bellow file

app\design\frontend\rwd\default\layout\wishlist.xml

OR

app\design\frontend\your_package\your_theme\layout\wishlist.xml

Find bellow code in above file

<block type="wishlist/customer_wishlist_button" name="customer.wishlist.button.share" template="wishlist/button/share.phtml" />

after comment like this way :

<!--<block type="wishlist/customer_wishlist_button" name="customer.wishlist.button.share" template="wishlist/button/share.phtml" />-->

First Way: Share Button coming in bellow file

app\design\frontend\rwd\default\layout\wishlist.xml

OR

app\design\frontend\your_package\your_theme\layout\wishlist.xml

Find bellow code in above file

<block type="wishlist/customer_wishlist_button" name="customer.wishlist.button.share" template="wishlist/button/share.phtml" />

after comment like this way :

<!--<block type="wishlist/customer_wishlist_button" name="customer.wishlist.button.share" template="wishlist/button/share.phtml" />-->

Second Way: Add bellow code in local.xml file (app\design\frontend\your_package\your_theme\layout\local.xml)

<?xml version="1.0"?>
<layout version="0.1.0">
    <wishlist_index_index translate="label">
        <reference name="customer.wishlist.buttons">
            <action method="unsetChild"><name>customer.wishlist.button.share</name></action>
        </reference>   
    </wishlist_index_index>
</layout>