Hello!
I've had several rounds of treatment with both donor eggs and donor sperm, and I'm currently almost 19 weeks pregnant following my most recent cycle.
I'd draw a distinction between 'donor embryos' - clinics that have frozen embryos available ready to use - and 'double donor' treatment where you use both an egg and a sperm donor but the two are brought together at the time of your treatment.
In the UK, the law requires that donor information be made available to any donor-conceived child once they turn 18, giving then the opportunity to potentially establish contact with their donors. In other countries the law can be very different - in the majority of the European destination countries for IVF, the law requires absolute anonymity. (There might theoretically be the possibility that family DNA services like Ancestry might one day help children in this circumstance trace donors, but of course it's hard to know how a donor who has signed up under the expectation of anonymity would respond to this.) Personally, it was a priority for me that any child have the opportunity to have this information in future, so I chose to stay in the UK for treatment. (It's also worth being aware that laws around who can access treatment vary wildly from country to country, so for example single people and same-sex couples may have a more limited range of options in terms of treatment location.) Treatment in Europe can be considerably cheaper in some circumstances, and is generally considered to be very modern and effective.
Donor embryos are relatively unusual to find in the UK (they're more common in the US, but there are often religious expectations around the recipients). They would generally be donated by couples who had completed their own treatment, so there would potentially be full genetic siblings somewhere out there. The advantage of using existing embryos is that, well, they exist! So you can be reasonably confident that you will get to transfer with a good-quality embryo, rather than having to wait to see how they fertilise and develop.
Some European clinics do have donor embryos available - it's worth checking the details of these. In some cases they essentially create double-donor embryos ready for use; in others they are the remaining embryos from other couples' treatment. In the latter case it's also worth checking whether they have genuinely been donated by the couples involved - I know of at least one clinic where they use embryos which couples have essentially left in storage but not come back for, rather than explicitly choosing to donate them.
If you choose to pursue double-donor treatment, you'll usually have a guaranteed minimum number of eggs available (sometimes with the possibility that the actual number could be higher). In the UK, one option is to use an egg-share donor - someone who needs IVF for whatever reason, but has good-quality eggs which they are willing to donate. In effect, you pay for (most or all of) their treatment, and they donate half their eggs (in some circumstances, you would get less than half, but there's a guaranteed minimum number). The waiting lists for egg-share donors can be shorter than for fresh eggs donated in other circumstances, and the aspect of helping someone else may also appeal to you, although it can be emotionally complicated if either the donor or recipient conceives and the other doesn't.
Fresh eggs are generally considered to have a higher success rate. The treatment involves synching up your cycle and the donor's, which can take a little time.
If you were to use frozen eggs, these can be available more quickly, and the protocol is more like an FET. With frozen eggs, you would also tend to have more choice of donors, since there isn't a waiting list involved in the same way.
Phew! I hope that helps a bit. Feel free to ask if I can be of more help!