My friend says there is a Rashi that says this, but I couldn't find it.
As Robev and Double AA noted, your friend was likely referencing this Gemara in Eruvin (100b)
אמר רבי יוחנן: אילמלא לא ניתנה תורה, היינו למידין צניעות מחתול - Similarly, Rabbi Yoḥanan said: Even if the Torah had not been given, we would nonetheless have learned modesty from the cat...
Where Rashi comments:
צניעות מחתול - שאינו מטיל רעי בפני אדם ומכסה צואתו - Modesty could be learned from cats since they don't expel their waste before people, and they cover their waste.
The Gemara Beitzah (25b) writes
אמר רבי שמעון בן לקיש, שלשה עזין הן: ישראל באומות, כלב בחיות, תרנגול בעופות. - Rabbi Shimon ben Lakish said: There are three brazen ones: The Jewish people among the nations; the dog among animals; and the rooster among birds.
Also, the Gemara Sanhedrin (108b) writes:
ת"ר שלשה שמשו בתיבה וכולם לקו כלב ועורב וחם כלב נקשר עורב רק חם לקה בעורו - The Sages taught: Three engaged in intercourse while in the ark, and all of them were punished for doing so. They are: The dog, and the raven, and Cham, son of Noah. The dog was punished in that it is bound...
Based on Chazal's descriptions, it appears dogs are among the most brazen and exhibit the least physical self-control, which is often at odds with the trait of modesty. (How Klal Yisroel is still modest yet brazen is a separate question.)