I'm bit lost. You have some app that wants to log in to Salesforce and pull images from it? Or do you want to call out from Salesforce and pull images from external app? For first one you likely need "connected app". But you need "remote site settings" or "named credentials" only for calling out.
I'm assuming it's option 1.
There are lots of ways to log in to SF, SOAP API (just username + password), REST API (lots of OAuth2 options, with username+pass or username + JWT or just OAuth2 client id and user logs in to SF interactively, you don't see the password)... Sounds like you're past this stage?
A successful login response will look bit like that (depends on method used):
{"id":"https://login.salesforce.com/id/00Dx0000000BV7z/005x00000012Q9P",
"issued_at":"1278448832702",
"instance_url":"https://yourInstance.salesforce.com/",
"signature":"0CmxinZir53Yex7nE0TD+zMpvIWYGb/bdJh6XfOH6EQ=",
"access_token":"00Dx0000000BV7z!AR8AQAxo9UfVkh8AlV0Gomt9Czx9LjHnSSpwBMmbRcgKFmxOtvxjTrKW19ye6PE3Ds1eQz3z8jr3W7_VbWmEu4Q8TVGSTHxs",
"token_type":"Bearer"}
You're supposed to take from it the access_token (that's your session id. It'll always start with org's id, compare with Setup -> Company information) and instance_url (that's where you're supposed to send any subsequent requests. No more calling the login gateways: login.salesforce.com, test.salesforce.com or mydomain.my.salesforce.com). If you're getting "Insufficient Privileges" I think you didn't change the endpoint.
You can test you logged in OK by sending a GET to the id endpoint you received. It'll give you OpenId info about your user.
Here's my GET to fetch OpenId data with the "Authorization: Bearer " request

So another GET with same header and I have my ugly mug:

If you want somebody else's picture - query similar to /services/data/v48.0/query?q=SELECT SmallPhotoUrl, FullPhotoUrl FROM User WHERE Id = '005...' should work. Check User fields.
{
"totalSize" : 1,
"done" : true,
"records" : [ {
"attributes" : {
"type" : "User",
"url" : "/services/data/v48.0/sobjects/User/(redacted)"
},
"SmallPhotoUrl" : "https://(redacted)/profilephoto/7293L0000008Tfq/T",
"FullPhotoUrl" : "https://(redacted)/profilephoto/7293L0000008Tfq/F"
} ]
}
The key thing is to use the new endpoint and pass the session id in the header. You might even find it easier to use Chatter API to pull photos (also REST-based)