Facebook Messenger: Right-click the sticker you want and click Copy Image, then paste it into your message.
Note: Currently, the web app messenger.com mattes the sticker against a black background instead of leaving it transparent. It works properly if you use the messenger popup in facebook.com though.
Other: Different apps and platforms behave differently; you may need to experiment to get them to display properly. Either
Copy and paste the url beneath the sticker you want to use, or
right-click the sticker to copy and paste the image itself.
(If you discover any tips for easily getting stickers to display properly on a specific app, let me know and I'll update this space!)