Sales Tax on Trades

Short answer? No.

Now, let's see why. Start here at the California Department of Tax and Fee Administration.

Your transaction is a barter: the action or system of exchanging goods or services without using money. According to my linked page,

The use of barter or exchanges is considered the same as making sales or purchases under the Sales and Use Tax Laws. The fair market value of the property or services received is normally the amount to which tax will apply.

That sounds like you should each pay tax, but this is not correct because the same page answers the question of what is taxable:

Retail sales of tangible personal property in California are generally subject to sales tax.

You and your neighbor are not engaging in a retail sale. It's a private exchange. No tax is due.

If either or both of you were in the business of selling your item (let's say you own a store that sells computers) then you would be responsible for paying sales tax by the barter rule, which prevents businesses from engaging in mutual trade without tax consequence.