*Yo ssi's Take*
In short what's going on: WhatsApp has updated their policies to allow them to share information with 3rd parties which can include advertisers etc.. while they won't be sharing your actual messages, they may share your phone number, app logs, status messages, profile name, profile picture, IP address and all transactions in their payment option they have in the works. Basically anything not E2E encrypted (see below).
Should you be worried?
Yes and no, depending on your level of comfort. I'd assume most facebook users are already having a significant amount of data shared.
Why are people moving?
Mostly because of the way WhatsApp is being run under facebook and where it may lead. It's definitely a good platform to be moving away from.
What is end to end (E2E) encryption?
It's a secure way to send messages without it being passed through or stored by a 3rd party. Something which isn't E2E encrypted your giving the option for the 3rd party's server to be compromised, and a potential point of entry, not to mention the 3rd party can access your data (WhatsApp messages calls and video calls are E2E encrypted). That's the reason at times you couldn't download that picture from a group because the picture has to be physically on the phone of the sender and isn't stored on the cloud.
What are the alternatives?
*Signal*
*Most closely resembles WhatsApp as it only sends E2E encrypted messages, but instead of being run by a big corporation, it's run by a non profit with security as it's focus.
The cons:
Having real E2E encryption means your data can't be backed up (to the cloud), because as soon as you back it up your relying on a third party to store the data and that ruins the encryption (So if you have your WhatsApps backed up to google drive or iCloud your basically loosing your E2E encryption). Signal "solves" this problem by only allowing you to preform a local backup which you have full control over (which uses your space on your hard drive)
*Telegram*
*A feature rich messaging app with lots of functionality and multiple advantages.
*One of the main features of Telegram is it's cloud storage, nothing has to be saved to your phone and everything is stored on telegrams servers. You can share files up 2GB(!) you can have all your pictures and videos taking up zero space on your phone, and as soon as you need it you can redownload it. It allows for a multi device login because everything is stored on the cloud.
So the obvious con with that is it's not E2E encrypted because that would make it loose all it's functionality. So Telegram has an additional style chat called secure chat. It allows for E2E messages which are only stored on one device and has all the advantages that come with E2E (think of the time you wanted to send someone a picture of an ID)
The cons:
Telegram allows searching globally for chats of all kind and therefore different companies have made versions of telegrams to solve those issues.
Notes:
Telegram regular chats are still secure, more details about that on their website. (The FBI tried to get in but haven't..)
WhatsApp has a status feature which I don't believe signal or Telegram have (it as well isn't E2E).