NOTE The following assumes a linear speedup (such as would be gained in a brute-force attack) - the speedup may be EXPONENTIALLY MORE, depending on the algorithm.
It is very difficult to comprehend big numbers - even I was surprised when the answer came up. Here is a way of thinking about it:
If, without any information, it would take 13.8 billion years (the age of the universe so far) to crack a key, with the help of 8 binary characters, it would take only 0.023 seconds.
This is: 13.8 billion years / 2(bits_per_symbol*number_of_symbols).
You said the number of symbols is "8 characters", and I'm assuming binary, which has 8 bits per symbol. so: 13.8 billion years / 2(8*8)
If they are uuencoded, or base64, they will have 6 bits per symbol, so it would take them all of 25 minutes to work through the remaining combinations.
These proportions apply only to the number of bits you've revealed, and is irrespective of how long the key is (only that the key would take 13.8 billion years to crack).
The whole point of public key cryptography is that you NEVER EVER EVER need to share the private key. Each private key should exist only in one place, and never travel across a network. Sending keys goes against the very principle of public key cryptography. There is NEVER a need to send private keys, partially or otherwise.
If you want two (or more) different devices to be able to decode the same message, have each create their own private key, send you their public key (securely!!) then encrypt the message with both keys. Usually with PKC, long messages are encrypted using symmetric encryption with a random key, and the key is then encrypted with PKC, and sent with the message; you can easily encrypt the random key with multiple public keys, and send all of them with the same message.
If all you want to do is show you have the private key, you can do the following:
Ask the person you want to prove it to to provide a random value (a "nonce"). Add random value of your own, hash it, and sign the hash. Send the your random value, and the signature, back.
Your counterpart will then take the nonce they sent, plus your random value, hash it, and check the signature against your public key.
Including a random value from the sender, proves that you did not just select a value that has been signed by the real owner previously.
DO NOT UNDER ANY CIRCUMSTANCES accept something sent by someone else, and signing it, without any changes. They can send you the fingerprint of a document they wrote, and you will be effectively signing that document.