What worked for me
import databricks.koalas as ks
df1k = df1.to_koalas()
df2k = df2.to_koalas()
df3k = df1k.merge(df2k, on=['col1', 'col2'])
df3 = df3k.to_spark()
All of the columns except for col1 and col2 had "_x" appended to their names if they had come from df1 and "_y" appended if they had come from df2, which is exactly what I needed.