Our friends and we often barter babysitting time with each other. Do we owe taxes?

I wouldn't worry about it.

This is from the IRS:Topic No. 420 Bartering Income

Bartering is the exchange of goods or services. A barter exchange is an organization whose members contract with each other (or with the barter exchange) to exchange property or services. The term doesn't include arrangements that provide solely for the informal exchange of similar services on a noncommercial basis (for example, a babysitting cooperative run by neighborhood parents). Usually there's no exchange of cash. An example of bartering is a plumber exchanging plumbing services for the dental services of a dentist.

The part I bolded explicitly says that a small informal babysitting group isn't a barter exchange.

That saves a lot of tax forms.

The Internet has provided a medium for new growth in the bartering industry. This growth prompts the following reminder: Barter exchanges are required to file Form 1099-B, Proceeds From Broker and Barter Exchange Transactions for all transactions unless an exception applies.

Publication 525, Taxable and Nontaxable Income

Discusses bartering but the examples have more to do with barter exchanges and small business.

Bartering

Bartering is an exchange of property or services. You must include in your income, at the time received, the FMV of property or services you receive in bartering. If you exchange services with another person and you both have agreed ahead of time on the value of the services, that value will be accepted as FMV unless the value can be shown to be otherwise.

Generally, you report this income on Schedule C (Form 1040). However, if the barter involves an exchange of something other than services, such as in Example 23, later, you may have to use another form or schedule instead.

Example 20.

You're a self-employed attorney who performs legal services for a client, a small corporation...

Example 21.

You're a self-employed accountant. You and a house painter are members of a barter club. ....

Example 22.

You're self-employed and a member of a barter club. ...

Example 23.

You own a small apartment building. In return for 6 months rent-free use of an apartment, an artist gives you a work of art she created....