There are two value sets here: yours and the prospect's.
You may value your service at $X, but if your prospect doesn't see it that way, that value does not exist.
However, you do have to start off with your target value. Have a "base/best case" value, and a "bare minimum/worst case" value.
The worst case value is the lowest trade price you will trade your services for. If the prospect values it at less than that, you won't trade.
Your trade prospect will probably have the same kind of numbers in mind for their own service, whether consciously or not.
When the two of you talk, you need to match up value. How much of your graphic design etc. do they want in return for some of their printing services? Money is the value medium. Think about it as two farmers trading cheese for chickens if that helps you.
After setting a qualified meeting, ie. they are interested in what you offer...
Begin by asking them how much they value your service at. Note we are getting at THEIR belief here, not yours. Because that's reality.
They will lowball, probably, but now you know the starting point.
Then ask them what they believe their service is valued at.
Now you have a basic idea of how much of your 'Service A' you need to provide in exchange for one unit of their 'Service B'.
Is this acceptable to you?
Maybe you found a really fair prospect and the numbers are great. In that case you can shake hands and sign the deal. Otherwise, it's time to negotiate.
If their valuation of your service was much lower than their value of what they provide, call them on it. "Why is your service worth so much to you, and mine so little?"
Always be ready to get up and leave. DON'T get emotionally invested and huff and puff if their valuation isn't to your liking. It's just business.
But the prospect will probably start waffling and making adjustments in your favor now.
If you just can't get close enough to that minimum figure to make things work for you, consider further trading. What can they give you that isn't a big deal for them but has great value to you, that will make up that difference for you?
Bartering is a huge part of the economy and large companies do it all the time. They try to hide this from consumers, because they want regular people to pay retail prices in cash. But up the chain, trade is commonplace.
Get some experience with this...keep at it and it will pay off.
Since you're looking to barter for printing services, the company your approaching will probably have their own prices advertised already, which simplifies one half of the discussion.
As for your end, if you can point to what services similar to yours cost elsewhere in the open market, point to that.
You can also ask your prospective client what they currently pay for such services. I'm guessing a printing company will have paid for graphic design in the past. So they ought to have numbers on file for both sides of the transaction.
There are no set rules on how to do this. Early on, many companies do this because they do not have the $$ to pay for services. I think doing this is smart, and is a true example of the Hustle.
Start with how much your service is per hour/gig/whatever metric you want to use. From there, be open to hearing what the similar business charges. Star there.
One of my favorite parts of bartering services is the potential to evaluate future partners with opposing skill sets. You get to evaluate many things from the timelines of the their work, quality, etc.
This is a broad question, so let me know if you would like to hop on a call to discuss this further. I would be happy to chat and tailor the answer more towards your specific business.