Gratis and/or open source web feeds for web and/or Android, separated according to whether they have or they don't have export and/or import capabilities, plus Thunderbird, which works on desktop.
The News app is an extension to Nextcloud. Essentially it is an RSS/Atom feed aggregator and offers a RESTful API for app developers. There are several clients, also for mobile devices.
Export: yes. Import: yes.
Ads: no. Trackers: ?
Android: https://play.google.com/store/apps/details?id=de.luhmer.owncloudnewsreader
NewsBlur is a personal news reader that brings people together to talk about the world. It shows you the original site and allows you to read stories directly off the site while keeping track of what you read. It also filters and highlights stories you like and dislike.
Export: yes. Import: yes.
Ads: no. Trackers: no.
newsrdr is a Web based RSS/Atom news reader that allows you to follow the latest news from your favorite websites in one place.
Export: yes. Import: yes. Source: https://github.com/tmiw/newsrdr
Ads: yes. Trackers: yes.
Organize and read content from RSS feeds into a magazine-like experience, simple sharing options, and seamless access across all devices.
Export: yes. Import: yes.
Ads: no. Trackers: yes. CDN: Cloudflare.
Feed Rinse is a web-based tool that allows you to filter unruly RSS subscriptions and block irrelevant posts.
Export: yes. Import: yes.
Ads: no. Trackers: yes.
A great alternative to 
Google Reader. Offers awesome productivity features like Search, RSS feeds for your starred items, tags & folder. You can share on 
X, .
Export: yes. Import: yes.
Ads: no. Trackers: yes. Hosting: Amazon.
Flym News Reader is a light, modern, totally free (no ads) and opensource project which keeps you inform by fetching your websites/blogs and displaying them in a mobile-optimized way.
Syncronization: no.
selfoss is a multipurpose RSS reader and feed aggregation web application. It allows you to easily follow updates from different websites, social networks and other platforms, all in a single place. It is written in PHP, allowing you to run it basically anywhere.
Requires self-hosted: yes.
Looking for a simple and open feed reader?
Free, open source, easy to use, Sismics Reader is what you need to keep your feeds under control.
Central
Access all your RSS and Atom feeds in one central easy readable place.
Easy
Sismics Reader can be installed in few steps on al.
Requires self-hosted: yes.
birdreader - Home-made, Open-source Google Reader replacement powered by 
Node.js and 
CouchDB (Cloudant).
REPORT: website not available. Requires self-hosted: yes.
FeedSpin is a RSS news reader that treats all sources equally, unless you say otherwise. So your must-read sources will always appear first. And any other sources will not be shown again until you have seen the updates of all the other ones.
REPORT: website not available.
Handy News Reader is a free no-ads RSS reader for reading offline contents from RSS feeds or web sites without RSS feeds.
Syncronization: no.
Cross-platform email client providing POP/IMAP support, robust filtering, multiple accounts, and chat. Extendable via add-ons with RSS/Atom, etc.
Syncronization: yes. Google account.
spaRSS is a light, modern, totally free (no ads) and open source project which keeps you inform by fetching your websites/blogs and displaying them in a mobile optimized way. It is based on the "Flym" RSS reader.
Syncronization: no.
Feeder is a fully free/libre feed reader. It supports all common feed formats, including JSONFeed. It doesn't track you. It doesn't require any setup. It doesn't even need you to create an account!
Syncronization: no.
Other services which can't import/export feeds but are still interesting and open source.
NewsTab is the ultimate newsreader for international readers, supporting top news sources in over 130 regions and languages, and syncs across all your devices.
Export: no. Import: yes.
The Old Reader is a web based RSS reader, something like Google Reader . Import existing subscriptions, find Facebook friends and more.
Export: no. Import: yes.
Aggregate RSS, newsletters, and social feeds with advanced filtering, real-time cross-device syncing, customizable reading views, and ad-free experience.
Export: no. Import: no.
CakeReader collects the content you care about. We help you discover quality content, subscribe and read with ease and organize everything in one place. You can add your own RSS feeds and subscribe to Youtube channels aswell to bring all your favorite content at one...
Export: no. Import: no.
TTRSS-Reader is a client application for the project 
Tiny Tiny RSS , a PHP-based online feedreader which runs on your own webspace. If you want to host your own feedreader, want a cross-device-solution and don't want to use...
Feedbin is a modern web reader to follow websites, Twitter feeds and email newsletters in one place. It boasts a clean interface, is ad-free, and costs $5/month.
Rivulet is a simple way to get content in one place from all over the web. Like Google Reader, it loads web feeds (RSS and Atom) and displays them to the user. But, Rivulet is simpler and easier to use.
An open source RSS news reader to help you discover, track and manage only the content you care about. Getting better every week and made with ? in NYC.
Displays a sidebar that lets you manage your RSS feeds in a folder tree view in Firefox 57+. A "Legacy" theme is available to have look close to "Sage".
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