What is the basis for eating in someone's house without a kashrut certificate?

No one asks for a kashrut certificate to eat in someone's house; they just make a rough judgement based on the host's reputation (the way they dress or behave, the circles they move in, etc.). That level of laxity would not be allowed in any other kashrut-related decision.

I had the funny idea for Hasgacha Pratit, a kashrut certification organization for private homes -- Don't even think of taking that seriously! But if you do, cut me in :-)

It is not just a matter of trust. You might ask your host "Do you keep kosher?" And the host might honestly say "Yes." Some Jews sincerely think that not having pork and shrimp in the house is enough for one to "keep kosher". Some follow ingredient labels and don't require hashgochas. Some Jews rely on hashgochas that other Jews reject.

You might have the chutzpah to pry a little, but lemayseh, few guests quiz their host on the details -- certainly not on the level they would demand for any other source of food. Usually, if the guest "acts frum", that's enough.

So: Why do people treat kashrut in other people's homes so leniently?

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Edit: Apparently it is more common than I believed for guests to query their hosts "What hashgochas do you accept? You do separate milk and meat dishes, right?", "Do you use meat-based gelatin?", etc.

I still think that such questions are not as widespread as one might think, and that general guests rely on vague impressions given by the clothing the host wears or the shul they belong to, but that goes a long way towards an answer.

Answer

The reason we require mashgiach and hechshers on products that you buy in the store or from a resteraunt, is because the person you are purchasing the food from gains a financial benefit from you purchasing. (obviously) There is therefore a temptation to lie or to cut corners in order to make a greater profit or to attract more customers.

However, the opposite is true when going over to a person's private home. People do not gain financially when you eat at their homes. In fact, they lose financially. So we assume there is no reason not to trust them.