The Life Cleric has 2 separate features which you've conflated here. The first one is Disciple of Life.

Whenever you use a spell of 1st level or higher to restore hit points to a creature, the creature regains additional hit points equal to 2 + the spell’s level.

The second one is Blessed Healer.

When you cast a spell of 1st level or higher that restores hit points to a creature other than you, you regain hit points equal to 2 + the spell’s level.

Blessed Healer is simple. You get healed when you cast a spell. So Regenerate will heal you for 9 hp, but only when you cast it.

Disciple of Life is a bit less clear. It is triggered "when you use a spell to restore hit points to a creature". There are 2 interpretations of this:

1. The first is that using the spell is the same thing as casting it, and therefore only the initial 4d8 + 15 gets the +9 hp from Disciple of Life.
2. The second is that you're using the spell to restore hit points every time it restores hit points. In that case, the target would indeed gain 10 hp every turn for the duration.

Unfortunately, there is basically nothing in the rules to help decide which interpretation is correct. "Using" a spell is not common terminology - almost everything in the rules talks about casting a spell, not using it.

However, a similar question was answered in the official Sage Advice Compendium.

If I’m a cleric/druid with the Disciple of Life feature, does the goodberry spell benefit from the feature? Yes. The Disciple of Life feature would make each berry restore 4 hit points, instead of 1, assuming you cast goodberry with a 1st-level spell slot.

If Disciple of Life works every time someone eats a goodberry, it should certainly work every time someone is healed by Regenerate. So yes, Regenerate cast by a Life Cleric will heal 10 hp every turn.

The intent is apparent from the first line of the spell description (NO!)

The first line does state (emphasis mine):

You sacrifice some of your health to mend another creature’s injuries.

This heavily implies that you are not eligible, but the mechanical description sentences that follow (see below) do not list a requirement such as that when actually choosing the recipient of the hit points.

Jeremy Crawford does reinforce the importance of that first line and that You should not heal yourself with Life Transference (emphasis mine.)

The first sentence of life transference expresses how the spell works: "You sacrifice some of your health to mend another creature's injuries." That's someone else getting healed, not you.

Physician, heal thyself?

The spell language seems to contradict itself. The second sentence describing the mechanics of the spell (Xanathar's, pp160) requires the following (emphasis mine):

You take 4d8 necrotic damage, and one creature of your choice that you can see within range regains a number of hit points equal to twice the necrotic damage you take.

The mechanical requirements per that sentence are that you take damage, and that you then choose one creature within 30' that you can see to regain hitpoints.

It does not state that it must be one other creature or that you can't choose yourself, so there really is no limitation here that precludes you taking the damage and receiving the hitpoints.

Self-Transference?

This is highly unlikely. Crawford has clarified that is not how the spell works, and this is most likely a problem with the wording of the spell in that second sentence.

We can take from the name Life Transference and the language of the first sentence to conclude that you are not supposed to be able to do this to yourself.