I would only offer what you think it is worth. Don't offer more than you think it's worth, or you'll regret that too.

Just mention to the people that show you around that you are really interested. Don't go overboard or you might put them off!! Put in an offer that you are happy with and wait. You might find that once the owners know that there are 2 people interested they might negotiate with both of you. When two parties were interested in a house we were selling the agent requested they put in their highest bid each. Worked out great for us (as the vendor) and they winning couple never knew they offered $24,000 more than the others, they just paid what they thought it was worth to them.