Edit: Thanks to Don Barry for pointing out WebRTC security issues. (2018-04)
Several protocols and programs are available for both public and private collaborations. It is useful to first think to a specific protocol, and then to look for a software implementing it. In general, several libre programs can be used to exchange information through a given protocol (proprietary programs instead rely on formats that are defective on purpose, e.g., lack of compatibility, presence of back-doors, etc.).
Hence, rather than creating a specific account valid only for the related service, users are allowed to create an account with a chosen provider, and then employ it across several different client programs (sometimes an account is not even required). Users who administrate their own servers are also allowed to install and configure server programs, without the need of relying on external services. For example, institutional servers can be used to host communication or file synchronization software, then available to internal or external users via client programs.
Collaboration tools for software development are not discussed here.
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XMPP (Jabber) is a widely used protocol for instant messaging (IM), which can be employed together with cryptographic protocol such as OTR (Off-the-Record Messaging).
IRC (Internet Relay Chat) is a protocol created in the late 80s by a Finnish developer, still very popular especially among collaborations involving open (but also private) discussions. It allows instant messaging over public or private channels hosted on servers.
SIP is a protocol allowing video chats, which can be used together with cryptographic protocols such as ZRTP (Z Real-time Transport Protocol).
WebRTC is an increasingly used standard that defines a collection of communications protocols enabling real-time communication over peer-to-peer connections, typically through application loadable with an Internet browser. WebRTC security alone is not as trustworthy as ZRTP verfication against man in the middle attacks, though.
Emails are nowadays still widely employed and unlikely to be replaced soon. It is worth recalling that messages are generally not encrypted. This is not an issue for public mailing lists. Additional tools are to be configured if private content has to be exchanged.
A wiki is a website allowing collaborative modifications of webpages through a web browser. Note that a wiki is a content management systems rather than a communication protocol. Libre wiki software usually offer standard interfaces and tools.
Video chat (ZRTP encryption) and IM (OTR encrpytion). Excellent cross-platform support, binaries are available for all main desktop and mobile OS. Good quality and reliable service. Support is given through mailing lists and IRC channel.
It may be problematic behind a VPN connection.
Jitsi client (and server) software is compatible with several instant messaging and voice/video call protocols. Furthermore, Jitsi provides a WebRTC-based software, Jitsi Meet, for encrypted video chats and IMs. Several tools are available such as screen sharing, documents sharing (through Etherpad), conference calls and live streaming. Conferences can be protected by a password.
No account nor client software are required to join Jitsi Meet. Everything is accessible through the web browser, hence it can be used from any OS. The server software can be installed to create and customize local instances. Public instances are available. Try the one maintained by the Jitsi Meet team.
Support is provided through mailing lists and IRC channel.
Reliable software, large community (mailing lists and IRC channels available), cross-platform desktop support. It can be used with many IM protocols and IRC. Supports file transfer. Several plugins are available. In particular, OTR message encryption can be set up and manage exchanges between different instances of the same account.
No mobile support yet.
Real-time collaborative editor. Public instances are available, and private instances can be installed and customized on private servers. Sessions can be protected by a password. Notebooks can be exported. Simple but reliable editor.
Cloud service (access and synchronization of files, contacts, calendars and bookmarks). Public instances are available. Client software supported by common desktop and mobile OS. Server software can be installed to create and customize private instances.
When using public instances, it should always be considered that what is referred to as "cloud" is nothing else than other people's computers.
Additional tested programs:
Non-tested software, but worth considering based on their descriptions:
More communication and project management software is listed under the FSD.