I have brought my own scent-free powdered laundry detergent to Italy for the last 8 or 9 trips --- even the Italian baby detergent and the hypo-allergenic detergents have dyes and perfumes in them. We have searched and searched, and you do not want to spend precious vacation time doing this.
Until last year, I could bring as much as I wanted. Now, it has to be 12 ounces or less and the TSA will probably want to look at it and maybe test it.
For 30 days, for just my own laundry (and I pack light so I wash a lot of times) I take about 8 ounces. Less than what fits in a snack-size ziploc bag.
For the smaller, high efficiency washing machines in Italy, I use a rounded tablespoon of the powdered detergent mixed in a cup or glass with a very small amount of hot water to turn it into something resembling a liquid laundry detergent. Their machines will generally not dissolve a powder.
You do not need as much detergent to wash clothes as many people think you do, and if you have allergies, you REALLY don't want to use too much or to have to put the clothes through a whole 'nother wash cycle just to double-rinse them (which I do at home, with white vinegar).
We try to remember to ask the host which compartment to put the detergent into --- it is not at all obvious, even if you can read Italian words. I would wait for the tour of the apartment rather than attempt this in an email.
Hope this helps! I now feel like a minor expert on this tiny subject!