Skype
Any card sent through Skype already has the functionality to be forwarded to other users. All the user has to do is press the forward button that displays on the side of the card. However, Skype does not support adding a share button directly on the card.

Facebook
You have to send a Messenger Template through the activity's channel data to include the share functionality on Facebook Messenger. In the template's element, you have to add a button with type element_share
. See the example below.
await turnContext.sendActivity({
channelData: {
"attachment": {
"type": "template",
"payload": {
"template_type": "generic",
"elements": [
{
"title": "Three Strategies for Finding Snow",
"subtitle": "How do you plan a ski trip to ensure the best conditions? You can think about a resort’s track record, or which have the best snow-making machines. Or you can gamble.",
"image_url": "https://static01.nyt.com/images/2019/02/10/travel/03update-snowfall2/03update-snowfall2-jumbo.jpg?quality=90&auto=webp",
"default_action": {
"type": "web_url",
"url": "https://www.nytimes.com/2019/02/08/travel/ski-resort-snow-conditions.html",
"messenger_extensions": false,
"webview_height_ratio": "tall"
},
"buttons": [{
"type":"element_share"
}]
}
]
}
}
}
});

Hope this helps!