While my friend was getting a tattoo the tattoo artist said you'll bleed more because you're drunk.
Is this true?
The answer seems to be probably yes.
Rats were given ethanol to a level consistent with human levels that would cause intoxication 
Bleeding time and blood loss were increased 1 h after ethanol administration, regardless of the utilized technique.
The Framingham Offspring Study looked at platelet function in 1-2k subjects, and found
Alcohol consumption is inversely associated with both platelet activation and aggregation, particularly in men. Additional research is needed to determine whether these findings contribute to the contrasting associations of alcohol consumption with risk of thrombotic and hemorrhagic cardiovascular events.
So, there is some experimental evidence to suggest that there is impaired platelet function after consumption of alcohol, and in those who consume it regularly. Platelets are involved in coagulation so bleeding times could be lengthened.
 Zoucas E, Bergqvist D, Göransson G, Bengmark S. Effect of acute ethanol intoxication on primary haemostasis, coagulation factors and fibrinolytic activity. Eur Surg Res. 1982;14(1):33-44. PubMed PMID: 7200888.
 Mukamal KJ, Massaro JM, Ault KA, [..], Tofler GH. Alcohol consumption and platelet activation and aggregation among women and men: the Framingham Offspring Study. Alcohol. Clin. Exp. Res. 2005 Oct;29(10):1906-12. PubMed PMID: 16269922.