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What kind of/How a secret language could be developed in middle ages by a secret society?

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1350 AD. A secret society is recruiting adepts in Rome, Florence, Paris, Wien, Prague and London; well-educated people in high ranks of nobility, catholic church, engineers, professors and knights. Given at that time Latin was used as formal/common language and cryptography was still relegated to military tasks and often required the burden of mechanical apparatus, even if encryption was limited to plain symbol rearrangement; how could a secret language be developed to let people talk each other, or exchange messages, while resembling a different but unknown language to other listeners/readers?

My requirements are: use of Latin letters; both cyphertext and plain-text are "readable", in a sense they would seem like natural language even if not understood (ie: Voynich manuscript); and people should be able to speak that language, not just reading/writing; no device should be needed for reading/listening, possibly the same also for writing/speaking.

This question could go to a more technical site, but I am open to fantasy answers, ie like using magic.

Voynich Manuscript

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I remember a book I read several years ago now, involving the Voynich manuscript. The basis was it was not decodable in any way - until a university student who happens to know symbols of another ancient language reads it, and finds two of these symbols. The passage between the two is the secret message, also written in this language.

Now, while that doesn't quite meet your requirements, it can be modified: instead of symbols you can just use standard letters but organise a start code (think DNA and start/stop codons). So, whenever a member of this organisation reads the letter sequence ABCDE, they would know that the message is between that and the next instance of it (or a different stop string, for example WXYZ).

Obviously, to increase the security you can use a basic cipher. Consider that at this time, basic ciphers of today were considered advanced, and while a basic Caesar cipher might be a little too basic, a Vigenére might meet the needs. If you want even more security, you can combine ideas from every answer: the start/stop strings, encipherment, and steganography (every $n + 2$ word).

In this way, messages can be distributed as text without fear of being read because you have to know several things before you can decode them. Let's try an example:

OJENF EIRYF WIRHUWBWUD WRIHWGWWD 
UEHYDYWBW **ABCDE** IFMMP ISRBWUR IPX 
WRIHGDIYWD BSF SUDB ZPV **WXYZ** HOEFF 
ISHFGIWOWDB IHSFWWODBD

Applying the techniques described, try to decode that. It's easy if you know, hard if you don't.

- ABCDE is the start sequence, WXYZ is stop
- Every second word within the block (starting with the first) is part of the message
- The cipher is a simple Caesar +1 cipher
- Encoded message: IFMMP IPX BSF ZPV
- Decoded message: HELLO HOW ARE YOU

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