Let me try to answer this part of your question:

> "Does it imply some specific condition on the maps $\phi$ and $\varphi$?"

**The quick answer:** $\phi$ and $\varphi$ have to be surjective submersions.

I recommend e.g. the paper [arXiv:0702399][1] by Blohmann for more details, but let me try to explain the basic idea here. The definition of Morita equivalence for Lie groupoids involves the concept of a *biprincipal bibundle*. We can break this down into the following notions:

 - *Groupoid actions.* Let $G\rightrightarrows G_0$ be a Lie groupoid, and let $l_X:X\to G_0$ be a smooth map. A *groupoid action* of $G$ on $X$ along $l_X$ is a smooth map
$$G\times_{G_0}^{\mathrm{src},l_X}X\longrightarrow X;\qquad (g,x)\longmapsto gx$$
satisfying some further properties that straightforwardly generalise the notion of a group action. Sometimes you'll see these written as $G\curvearrowright^{l_X} X$. Actions from the right are defined analogously.
 - *Groupoid bundles.* This is a notion that will eventually allow us to generalise Lie group principal bundles. A $G$-*bundle* is a smooth map $\pi:X\to B$ that is invariant under some action $G\curvearrowright^{l_X} X$. You might write this as $G\curvearrowright^{l_X} X\xrightarrow{\pi}B$.
 - *Principality.* A Lie groupoid bundle $G\curvearrowright^{l_X} X\xrightarrow{\pi}B$ is called *principal* if $\pi$ is a surjective submersion, and the $G$-action is free and transitive on the $\pi$-fibres.
 - *Bibundles.* A bibundle between two Lie groupoids $G\rightrightarrows G_0$ and $H\rightrightarrows H_0$ is a pair of actions $G\curvearrowright^{l_X}X$ and $X{~}^{r_X}\curvearrowleft H$ that interact well together; in particular we must have two groupoid bundles $G\curvearrowright^{l_X} X\xrightarrow{r_X}H_0$ and $G_0\xleftarrow{l_X}X{~}^{r_X}\curvearrowleft H$. Here the smooth maps $l_X$ and $r_X$ are the arrows $\phi$ and $\varphi$ from your question, respectively.
 - *Biprincipality.* A bibundle is called *biprincipal* if both the groupoid bundles described in the previous bullet point are themselves principal groupoid bundles. From the definition of principality for groupoid bundles it follows that $l_X$ and $r_X$ (i.e. $\phi$ and $\varphi$) have to be surjective submersions.

Also note that there is a relation between the formalism of (biprincipal) bibundles and Morita morphisms (also known as *weak equivalences*). It is known in the literature that (one-sided) principality of bibundles corresponds to essential surjectivity and full faithfulness of Morita morphisms.

  [1]: https://arxiv.org/abs/math/0702399