By an algebraic theory $\mathcal{T}$ I mean any category with finite products such that the objects are given by all finite powers of some object $X$. Let $Alg\mathcal{T}$ be the concrete category of finite-product-preserving functors $\mathcal{T} \to Set$, equipped with the obvious forgetful functor (this is Adamek et al.'s definition for the category of algebras over $\mathcal{T}$ and is, according to them, equivalent to Lawvere's original definition).
To an algebraic theory $\mathcal{T}$ in the above sense corresponds canonically an algebraic theory in the sense of model theory (i.e. first order and no quantifiers, relations, logical connectives) such that $Alg\mathcal{T}$ is “the” category of models of $\mathcal{T}$.
Now given two algebraic theories $\mathcal{T}_1$ and $\mathcal{T_2}$ such that $Alg\mathcal{T}_1$ is concretely equivalent to $Alg\mathcal{T}_2$, does this imply that the corresponding theories (in model theory) are bi-interpretable?
More generally I am wondering in how far information is lost when passing from an algebraic theory in the sense of model theory to the category $Alg\mathcal{T}$ (considered up to concrete equivalence), where $\mathcal{T}$ is the corresponding algebraic theory in the above sense. I would be grateful for ideas and/or references.
Yes. We can recover the category $\mathcal{T}$ from the concrete category $(C,U)$, where $C = \text{Alg}(\mathcal{T})$ and $U\colon C\to \text{Set}$ is the forgetful functor. The point is that the $n$-ary operations (arrows $X^n\to X$ in $\mathcal{T}$) correspond exactly to the natural transformations $U^n\to U$, i.e. $\mathcal{T}$ is the full subcategory of $\text{Set}^C$ on the finite powers of $U$. For details, see the section "A reconstruction theorem" in this blog post.
So if $(\text{Alg}(\mathcal{T}_1),U_1)$ is concretely equivalent to $(\text{Alg}(\mathcal{T}_2),U_2)$, then $\mathcal{T}_1 = \mathcal{T}_2$. I gather from your question that you already know that an algebraic theory $\mathcal{T}$ determines a model-theoretic theory $T$ up to bi-interpretability.
Stronger, if model-theoretic theories $T_1$ and $T_2$ induce the same Lawvere theory, then $T_1$ and $T_2$ are interdefinable: each symbol in the language of $T_2$ is definable in the language of $T_1$, so that each model of $T_1$ determines a model of $T_2$ with the same underlying set (and vice versa). In a general bi-interpretation, we're allowed to specify that the underlying set of a model of $T_2$ is a quotient of a definable subset of a model of $T_1$ by a definable equivalence relation (and vice versa). It's useful to make a distinction between these notions.
You may also be interested in this MO question, about categories of the form $\text{Alg}(\mathcal{T})$, given without forgetful functors.